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. Judas' death
1. Other ideas about Judas
2. The field's buyer
1. When was the field bought?
3. Origin of the name Field of Blood

1. How did Judas die?

Luke's Greek does not say "falling headlong"; it says "became face-down" (prenes). 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 us 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.

This hypothesis 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".

Perhaps Matthew omitted the main tradition of Judas' rise to fame to focus on the divine reason of the field's name (Matt. 27:6-10). Such a theological motive is typical for him. The explanation is most likely Matthew's own, and this he does this type of pesher throughout his Gospel.

Haenchen maintains that the word chorion in Acts 1:18 implies a farm. And that Judas fell from the roof of the farmhouse. How exactly someone's guts "burst out in the middle" from a roof fall is a mystery to me, but 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. 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" - 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 this name? 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).

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!).

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. the name "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?

Did Judas buy the field himself (Acts 1:18) or was it the priests (Matthew 27:6-7)?

The Talmud relates that 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] 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 perhaps while Judas technically did not buy the field with the money, it was his money and in a sense ultimately he bought the field through his actions.

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.

Similarly, an inscription by Xerxes I on some column bases from Susa says,

King Xerxes says: By the grace of Ahuramazda, king Darius, my father, built this palace. - See here

Another inscription on a slab of marble from Susa says,

I am Xerxes, the great king, king of kings, king of all nations, the son of king Darius, the Achaemenid. King Xerxes says: I built this palace after I became king. This I ask as a favor from Ahuramazda: May Ahuramazda and the gods protect me, my kingdom, and what has been built by me. - See here

And then we have here,

A great god is Ahuramazda, who created this world, who created yonder sky, who created mankind, who created happiness for mankind, who made Xerxes king. One king for many, one leader of many. I am Xerxes, the great king, the king of kings, the king all kinds of people with all kinds of origins, king of this world great and wide, the son of king Darius, the Achaemenid.

The great king Xerxes says: By the grace of Ahuramazda, my father, king Darius built this palace.

May Ahuramazda together with the gods protect me, and what I built, and what was built by my father, king Darius. May Ahuramazda and the other gods protect this.

Xerxes greatly expanded Darius' palace at Susa as the last inscription states. The second inscription has permissible exaggeration since Xerxes did a lot of construction on the palace, whereas the first notes his father built the original. We see one could refer to the main cause behind something, as if that party directly did it all. Similarly, Judas was responsible for the field to be ultimately bought. Xerxes didn't decide to lie in his later inscriptions saying he built the palace, because his last inscription (XPc) wouldn't say "Darius built the palace" and that Xerxes built part of it in separate places. Moreover, Darius building the original palace would've been within living memory.

When was the field bought?

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). No matter where you place the sentence, "Judas bought a field", it will always seem prior to his death on it, even if one assumes it's a metaphor: "Judas died. Judas also bought a field." The metaphorical language would require to put it before the statement about death regardless.

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).

References

1. Papias Fr. 4 (Holmes)
2. 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
3. Josephus considers the suicide of a captured Judean who bashed his head into a wall courageous
4. Haenchen, Ernst. The Acts of the Apostles (English tr. of the German 14th ed. (1965), Westminster Press: Philadelphia (1971)), p.161
5. 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)
6. Yoma 84b: doing work to save a Gentile on the Sabbath is a violation, whereas for a Jew it's not
7. Haenchen, p.160, n.7
8. Haenchen, p.160, n.5