Was there a census under Quirinius when Jesus was born? Luke 2 vs Matthew 2

The passages

And it came to pass in those days, that there went out a decree from Caesar Augustus, that all the world should be taxed. (And this taxing was first made when Cyrenius was governor of Syria.)

Luke 2:1-2

Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king.

Matthew 2:1

The problem

Luke says the birth took place during a census when Quirinius (Cyrenius in the KJV) governed Syria. Josephus records a census of Judea under Quirinius, and dates it to the year when Archelaus was removed, which is AD 6. That census was a local assessment after Judea became a Roman province, and it triggered a revolt led by Judas of Galilee. Luke himself says Jesus was born in the days of Herod (Luke 1:5), and Herod died about 4 BC. That leaves a gap of ten years or so.

Other points of friction: no record outside Luke describes an empire-wide census under Augustus at that time, and a Roman census would not ordinarily require people to return to an ancestral town.

What believers and scholars say

A first census, earlier. Some read Luke 2:2 as "this was the census before Quirinius was governor", taking the Greek word for "first" as "before". This is grammatically possible and argued seriously, but many Greek scholars find it strained.

An earlier role for Quirinius. An inscription once thought to refer to Quirinius has been used to argue that he held an earlier command in the region around 6 BC. The inscription is fragmentary and its attribution is disputed.

A mistake by Luke. Fitzmyer, Brown and most critical scholars conclude that Luke confused the later census with the birth, perhaps because the AD 6 census was the best-known one in Judea. Luke wrote decades later and for a Gentile audience.

Where this leaves it

The attempts to save Luke's precision rest on one disputed reading of the Greek and one disputed inscription. The alternative is that a careful and often accurate historian got this one wrong. Both options are reasonable; neither has the force of a clear external record. This is among the most-discussed dating problems in the New Testament and no single answer has won general agreement.