Matthew 27:1-10 and Acts 1:18-19 seem irreconcilable on Judas' end. These two passages led C.S. Lewis to abandon inerrancy. A closer look turns much of the differences into complements.

The differences:

1. How did Judas die?
1. Prenes Genomenos
2. Farmhouse?
3. Other Ideas
4. Why Rehabilitate Judas?
5. Summary
2. Who Bought the Field?
1. When Was the Field Bought?
3. The name Field of Blood

1. How did Judas die?

Matthew's account says he hung himself. Acts says his guts burst. The classic solution since at least John Chrysostom has been that the rope snapped and while bloated, he fell and burst; or perhaps after some time on the ground.

Prenes Genomenos

Luke's Greek does not say "falling headlong"; it says "becoming face-down" (prenes genomenos). Prenes means "face-down" in Homer (Iliad 24.11) and medical texts like Hippocrates. In Homer, "falling face-down" occurs when a warrior has been killed. The same for jumping off a cliff in Greek plays. But the word for falling used is "pipto" or a variant (e.g. katapipto). Luke certainly knew this verb (prospipto, empipto, apopipto) and used it where "falling" is concerned, whether before someone's feet (Luke 5:8, 5:12, 8:28, Acts 16:29) or a freefall (Luke 6:39, 8:5, 10:18; Acts 5:5, 5:10, 9:4, 9:18, 20:9, 28:6). Many times this is not a quote and is in Luke's own language, such as the boy falling from the third story (Acts 20:9; see also Acts 9:4, 28:6). Would Luke really use the unambiguous pipto there, but give a description of how someone falling looks (they "become headlong") in Acts 1:18?

The Latin translation of Wisdom 4:19 which renders prenes as disrumpet - "burst" or "dash" (RSV) - was probably influenced by Acts 1:18, which the Vulgate also has as "burst":

And he indeed hath possessed a field of the reward of iniquity, and being hanged, burst asunder in the midst and all his bowels gushed out.

Without a doubt the resolution of Matthew and Acts was in the mind of the author there: he "possessed" not "bought" a field, and the insertion "he was hanged". The normal meaning of prenes would translate Wisdom 4:19 roughly as, "and He will dash them speechless prostrate", instead of the Vulgate's, "and He will burst them puffed up and shake them speechless". It seems that "puffed up" (inflatos) is an artificial insertion to render prenes as "bursting". This is confirmed by the secondary addition of "burst" in addition to "swell" that has to be added to the connotation of prenes.

Readings found in later Syriac, Georgian, and Armenian manuscripts that have presthes genomenos (became swollen) over prenes genomenos undoubtedly exercised the same kind of harmonization we see in the Vulgate.

In Iliad 23.25, 24.18 the slain Hector lies face-down (prenes) in the dust. This is probably what Luke had in mind: Judas was lying face down (obviously dead) and "burst in the middle and his guts poured out". How he got on the ground (dead), Luke does not tell us, and he may not have known. But to describe someone as prostrate or prone with his intestines spilled out is much closer to rotting (after a hanging or any other death), than it is falling from somewhere.

The similar expression of falling (not becoming) “face up” and turning someone on the ground from being “face down” is a Jewish method of stoning. [Mishnah Sanhedrin 6.4] This again suggests Luke maybe had Semitic (early) sources.

This idea is supported by Papias (c.120 AD), who reports a similar, though very exaggerated tradition:

Judas walked about in this world a sad example of impiety; for his body having swollen to such an extent that he could not pass where a chariot could pass easily, he was crushed by the chariot, so that his bowels gushed out.

Owing to the "gushed out" bowels, Papias' account is definitely influenced by Acts, but not Matthew. The fact that his tradition sought an explanation for the gut-bursting shows that Acts 1:18's "prenes genomenos" was never understood as "falling headlong".

The suggestion that perhaps Matthew omitted the main tradition of Judas' end to focus on his hanging in related to Dt 21:23 is possible. He wouldn’t have invented this as it was applicable to Jesus (Gal 3:13; Acts 5:30, 10:39, 13:29; M.Sanh. 6.4; cf. apocrypha where Peter requests an upside down crucifixion, being unworthy of Jesus’ death).

Papias’ sources knew Acts’ tradition so it must’ve been widespread. Thus MT likely knew it too, but chose a theological reason as he does elsewhere (eg the 2 donkeys). Possibly the detail of the hanging was less well-known. Alternatively, the longer quote from Papias says someone cut the rope before Judas died from the hanging. [Holmes (1992). The Apostolic Fathers, 323 (Fr.18)]

Regardless of how Papias reasoned about Judas' hanging, if he did at all, it seems the tradition he had was dependent on Acts, because it's late (exaggerating the swelling) and uses the word prestheis ("swollen"), which is the same apologetic some manuscripts tried by changing prenes. But unlike those manuscripts, Judas doesn't fall anywhere but lives swollen until he dies either from a chariot or his condition. This confirms again that someone in the early Church, whether Acts' source or someone reading Acts, did not see prenes genomenos as a fall or his death resulting from one.

Farmhouse?

Haenchen maintains that the word chorion in Acts 1:18 implies a farm. And that Judas fell from the roof of the farmhouse. A fall from the equivalent of a roof’s height would not have spilled guts and in fact frequently didn't even kill a person. [Mishnah Sanhedrin 6.4] And overall his claim is untrue. Matthew 26:36 and Mark 14:32 refer to Gethsemane as a chorion and the Greek of Acts 4:34 distinguishes between those owning land/field (chorion) and others who might own houses - "all those who owned lands or houses sold them" (HCSB). The Aramaic "haqel" is a direct rendering of the Hebrew "sadeh" which is best understood as a generic "field".

Another attempt by Haenchen to support the "farm" idea is that when Acts 1:20 says, "...Let his dwelling become desolate; let no one live in it..." it's talking about Judas' farmhouse being desolate, and cannot be about his place as an apostle because that was replaced. True it’s not about his apostolic position. But even if we assume the dwelling can't be his hometown's house, this is simply not true again. The desolation of his "dwelling" (epaulis) has a parallel in the Talmud, where "a corpse with no one to bury it acquires its place and is buried where it was found". [Bava Kamma 81a.2] The word "place" (maqom) can be used to imply dwelling (Gen. 18:33, 22:3; compare 1 Ki 8:13). Acts 1:20 is, in any case, simply quoting the Septuagint of Psalm 69:25 (LXX = Ps 68:26) which has epaulis. This is not a word Luke inserted.

Other Ideas

An attempt by some older commentators was to interpret apenksato in Matt. 27:5 as Judas "left choking up" i.e. emotionally. This has to be rejected seeing the usage in Josephus means hanging or strangulation.

Another idea sees Judas Iscariot to mean Judas who has askara - a serious, sometimes fatal disease that strangled. [Berakhot 8a.13; Shabbat 33b.2-4] This also has to be rejected: many had this and it would not have become a nickname for Judas. Nor did it become a posthumous nickname, if suggested, for his having hanged himself, because John 6:71, 13:2, 26 understands it as a family pedigree. So the common explanation "ish Kerioth" -> "Iscarioth" (Greek has no "sh" sound or letter) - man of Kerioth, a town 25 or so miles south of Jerusalem - is probably the case. This means he also wasn't a "Sicarii".

But did Matthew and Acts (!) form legends about Judas' death out of "Iscarioth"? The disease askara began in the intestines and traveled to the mouth, suffocating! [Shabbat 33b.2] Yet, this couldn't have been the case at least for Acts: a common disease that did not cause death from intestines (compare Agrippa's death in Acts 12:23). [Berakhot 8a.13; Shabbat 33b.2] The numerous cases of intestinal death in Josephus, two of which had worms, one of which Acts 12:23 repeats, is different from what happens in Acts 1:18. Guts don't just explode. And as noted, Judas was likely already dead in that verse.

So did Matthew imagine (or artistically create?) a hanging based on this? But Matthew knows that this is a noun as a place of provenance: he changes Mark 3:19 from "Iscarioth" (probably the original) to "Iscariotis" in Matt. 10:4 (and in 26:4) meaning Judas the Iscariote. Luke doesn't do this in the list of apostles (Lk 6:16). The tradition itself knows this is the "Iscariot man" and not the "hanged man" because Luke 22:3 has Matthew's form, as does John 12:4, 14:22. This means the word was probably never associated with askara either by Matthew or his tradition, or it would be Iskaris/Askaris. The inclusion of "theta" for "Iscarioth" is simply inexplicable if askara had anything to do with the name.

In addition, much of Matthew 27:1-10 has older Semitic roots. "haima athoos" - innocent blood - in Matt. 27:4 is a Hebraism (Deut. 27:25). The word can also be used simply for "innocent" (Num. 32:22). Did Matthew "rejudaize", having read the Septuagint? He certainly reuses the word "athoos" in Pilate's speech (Matt. 27:24). But there's nothing wrong with portraying Pilate's statements in Matthew's own words. And the manuscripts that change both 27:4 and 27:24 from "innocent" to "righteous" (dikaion) may make "rejudaizing" doubtful. He also knows the temple gift as "korban" (Matt. 27:6; see Mark 7:11). Acts 1:18's source is also Semitic: knowing the Aramaic for "field of blood".

Why Rehabilitate Judas?

The argument that Matthew wouldn't have invented a contrite Judas is not as strong as it might first appear. In a big twist of irony, Adam Clarke defends Jesus' innocence by pointing out that his betrayer willingly grieved and tried to undo his actions. Of course, the natural question is, did Matthew invent this for his own apologetic?

I agree with Adam Clarke that Matt. 26:24 does not constitute evidence for Judas' eternal damnation, based on similar conditional statements one could repent of in the Talmud. But even if we accept that hanging himself out of guilt perhaps did not constitute a sin, there is clearly no special preference for Judas in Matthew's Gospel. He preserves the name "Thaddaeus" for the other apostle named Judas (Matt. 10:3; Mk 3:18), which Luke changes back (Lk 6:16; see Jn 14:22). Matt. 26:24 does not soften the woe in Mark 14:21 the way Luke 22:22 does. This suggests no redemption.

Had Matthew wanted to defend Jesus and his church, he would've slandered Judas. The fact that out of all the Gospels, only John does, merely calling him a thief means there was never any pressing need to vindicate Jesus' betrayal on logical grounds. The worst insult he even receives in early Christian tradition is the "sad example of impiety" in Papias! A contrived change of mind would make sense only for a sympathetic character, or a villain who admits his error after excessive punishment (Agrippa I in Josephus - note Luke's omission of Agrippa's penitence in Acts 12:23; Antiochus IV in 2 Maccabees 9:9-13). If at least Matthew had something gruesome like in Acts instead of the typical suicide (hanging or falling on a sword), one might have a case:

In early Christian literature the death of Judas is broadly understood as a fitting end to the life of the betrayer of Jesus. Papias’ description of Judas’ death can be illuminated by comparison with ancient biographical and medical literature, in which oedema and parasitic infections are a consequence of greed, and also apocalyptic texts, in which worms become an emblematic form of divine punishment after death. Viewed in this context the death of Judas serves a pedagogical function as a warning about the dangers of greed. [Abstract to Moss, Candida R. "A Note on the Death of Judas in Papias", New Testament Studies 65.3 (Jul 2019)]

Matthew's disinterest in villifying Judas is clear when compared to later Christian views. In Papias he dies by a chariot when swollen. In later commentaries, starting in the 11th century (Theophylact of Ohrid), he becomes so grotesquely swollen, that he couldn't see and urinates worms eventually dying so that the stench remained there perpetually - clearly a legend. Paul Middleton even considers Matthew's account to be favorable as a "noble death". [Middleton, Paul. "The “Noble Death” of Judas Iscariot: A Reconsideration of Suicide in the Bible and Early Christianity", Journal of Religion and Violence 6.2 (2018): 245-66] Though it's clear Matthew is disinterested in rehabilitating Judas as discussed above, so perhaps we have a dispassionate and thus historical account by him instead.

Nor was Judas' death in Acts modeled on 2 Maccabees 9 with Antiochus Epiphanes IV. There's no "worms" involved (cf. Acts 12:23), and the death is completely different even if it depicts intestines. Antiochus' death makes more sense with Matthew's penitence, but we don't see anything but a simple admission of guilt by Judas (compare 2 Macc. 9:17).

Another, more complicated supposition is that the respective traditions of Matthew and Acts grew out of reflections upon the Old Testament. Haenchen refers to Schweizer in considering that Matthew's version was influenced by Zechariah 11:12 and Jeremiah 18:2-3, 19:1-2, 32:6-15, whereas the tradition cited in Papias comes from a reflection upon the water curse of Numbers 5:21-22, Psalm 69:24 and Psalm 109:18. In other words the various stories were invented out of the Old Testament.

But Psalm 69:24-25 says nothing about a "water curse", only a general punishment. Numbers 5:21-22 speaks of a punishment on an adulterous, pregnant woman and the curse doesn't threaten her physical life, only a miscarriage and infertility. Psalm 109:18 is one of the many types of curses to be found in the Old Testament. So for these random curses to have modeled Judas' fate is just as random as any other hypothesis. Their obscure attempted connection speaks against it.

Why would Matthew invent Judas throwing the money back into the temple (Zech 11:13!), when in Zechariah it is the righteous man who is persecuted that does this? One can even wonder how Jeremiah 32:6-15 could have influenced a connection to Judas, especially since the text is a signal of a restoration (vv.15, 42-44; cf. Acts 1:20!). Not to mention Jer. 32:15 is the exact opposite of the infamy of Akeldama.

Summary

The Greek phrase "prenes genomenos" in Acts 1:18 is best understood as "being facedown" (i.e. lying dead on the ground), with no cause of death given. Luke would not have described a fall in such a strange way without using the typical word, pipto, as he does elsewhere (Acts 20:9, etc).

Several things are shared by Matthew and Acts:

1. "Field of Blood" (Matt. 27:8; Acts 1:19)
2. Judas' death in Jerusalem
3. His death shortly after the betrayal

The statement that the field became a cemetery for foreigners (Matt. 27:8) seems to suggest a connection with the spilled guts in Acts 1:18: no Jew would've wanted risking ritual impurity by the accidental moving (while digging = carrying) or touching of Judas' entrails. [Mishnah Oholot 1.6] And there was no special love for Gentiles. Nor would the reason for a Gentile cemetery be because of the blood money, as if Jews would be unclean burying their dead, since they would've never been able to use the money in the first place.

Even if one ignores the possible application of Deut. 23:18-9 for money obtained through murder (e.g. money from idolatry was forbidden), the money Judas wanted to return was payment for murder as far as he was concerned, even if the priests didn't view it that way. [Zevachim 2a.1, 7] One cannot use money or offerings obtained through serious enough transgressions. [Sukkah 30a.2] Hence why it couldn't go into the Temple treasury. The priests would've preferred to repurpose it to something matching its origin. [compare Makkot 12b.10-13a.1] A field purchase with blood money would not have been an issue for Jewish burial, as the money was voluntarily returned by Judas, not stolen by the priests, so they could reuse it. So they "conferred" (Matt. 27:7) and Akeldama probably seemed like a good idea.

2. Who Bought the Field?

Judas (Acts 1:18) or the priests (Matthew 27:6-7)?

Acts or its sources abbreviate and do this elsewhere: not wanting to ruin the flow similar to how it says the unbelieving Jews crucified and buried Jesus in a tomb (Acts 13:29). Similarly, compare how "Levi" tithed Melchizedek through Abraham, by being an unborn descendant of his (Hebrews 7:9-10). So the priests may have bought it in Judas' name in a way due to blood money. [Bava Metzia 2a.5-6] So, the field was bought through Judas' actions and money, but technically by the priests.

A corpse without relatives or acquaintances to bury it that generated publicity "acquires" the location it's found at. [Sanhedrin 47b.14]

The Talmud says the "agent of a man is like the man himself." [Kiddushin 41b, Berakhot 34b, Bava Kamma 70b] In the tannaitic period (before 200 AD), the rabbis ruled that a man can, colloquially speaking, "find" something simply by seeing it, without the otherwise needed physical touch for it to be his. [Bava Metzia 2a.6-8]

That Luke-Acts was not written by a Semite is irrelevant: his sources are frequently Semitic, such as the infancy. And we see this here with his knowledge of the Aramaic name, Akeldama.

When Was the Field Bought?

Seeing the met mitzva (notorious corpse) was infamous, [Sanh. 47b.14] the priests probably bought the location after Judas' death and made it a burial place for foreigners seeing its negative perception and defiled condition. They had the power to buy and assign types of graveyards so. [Mishnah Sanh. 6.5]

Matthew implies the Pharisees bought the field after Judas hung himself. This isn't stated, but it's an easy decision with that unpopular and devalued land. Unless one supposes that Judas hung himself out of spite on the field they bought - not an unreasonable hypothesis (Matt. 27:4-5a) - it seems they bought the field after some consideration and subsequent to Judas' hanging.

Nor does Acts have to be following a strict chronology if we suppose some metaphoric language as discussed above. For example, he abbreviates the death of Agrippa I as if immediate (Acts 12:23).

3. The name Field of Blood

Matthew neither states nor implies Judas had anything direct to do with the field. But for everyone to know the place as "the Field of Blood" because of a blood money purchase (Matt. 27:6-8) is unlikely. Even if the priests told the seller, why and how would this spread the news far and wide to Jews who, especially in Jerusalem, were mostly not followers of Jesus (the frustration with Jerusalem - Matt. 23:37, Lk 13:34)? A cemetery for foreigners wouldn't create such a name.

So here we see Matthew's typical midrash. He is giving the theological reason: the why, not how, of the name's genesis - namely God's decision by the circumstances of its purchase. This is similar to king Jehoshaphat's plea for help:

Then the king of Israel and Judah’s King Jehoshaphat went up to Ramoth-gilead. But the king of Israel said to Jehoshaphat, “I will disguise myself and go into battle, but you wear your royal attire.” So the king of Israel disguised himself, and they went into battle.

Now the king of Aram had ordered his chariot commanders, “Do not fight with anyone, small or great, except the king of Israel.”

When the chariot commanders saw Jehoshaphat, they shouted, “He must be the king of Israel!” So they turned to attack him, but Jehoshaphat cried out and the Lord helped him. God drew them away from him. When the chariot commanders saw that he was not the king of Israel, they turned back from pursuing him. - 2 Chronicles 18:28-32 (HCSB)

The superficial or direct reason the charioteers ended their pursuit was recognizing this was the wrong king (2 Chron. 18:32). However, ultimately it was through God's intervention that Jehoshaphat was saved (2 Chron. 18:31).

Something similar is found in John 18:31-2, 19:24: neither the priests were consciously trying to fulfil Jesus' words, nor the Roman soldiers directly wanting to fulfil Jewish law.

References

1. Papias Fr. 3 (Donaldson Ante Nicene Fathers Vol 1)
2. Haenchen, Ernst. The Acts of the Apostles (English tr. of the German 14th ed. (1965), Westminster Press: Philadelphia (1971)), p.160, n.7
3. Haenchen, p.161
4. So Adam Clarke and Wakefield; although in his commentary on Matthew 27 Adam Clarke says that, "the old method of reconciling the two accounts appears to me quite plausible", he seems to have changed his mind in Acts 1
5. Josephus considers the suicide of a captured Judean who bashed his head into a wall courageous
6. The longer fragment of Papias on Judas' death is probably expanded, perhaps by Theophylact of Ohrid in the 11th century, whose source, Oecumenius, had the shorter version. [Routh (1846). Reliquiae Sacrae Vol. I, 26-7] Its dependence on Acts is obvious: uses chorion for the place of Judas' death. And his head is more swollen than a chariot?! Even as a metaphor, for a doctor to not be able to see his eyes doesn't make much sense. This longer version has Judas be swollen to grotesque proportions, with worms and pus after urination, and in the end dies with a stench that lasted to Papias' day. [Holmes (1992). The Apostolic Fathers Fragment 18]
7. Haenchen, p.160, n.5
8. Hardly is it implied in Matt. 27:5 Judas would've traveled far in his distraught state to commit suicide. Similarly, in Acts this became known "throughout Jerusalem", not Judea etc (compare Acts 9:42)
9. Yoma 84b: doing work to save a Gentile on the Sabbath is a violation, whereas for a Jew it's not