The Birth Stories in Matthew & Luke Are Not Compatible.

Timeline of the Birth Narratives from "The Mission: Part Two"

The birth stories in the Gospel of Matthew and the Gospel of Luke are not compatible.

The events in the story in the Gospel of Matthew are pretty well wrapped up before the events in the Gospel of Luke even begin.

It's one thing to have the information and thank God for the age of the internet for that, but it's another thing to see it visualized.

Yet, there will be those who refuse to accept it, perhaps even fabricate a Roman census during the reign of Herod the Great, which would have been utterly unnecessary and out of place and that just so happens to use the name from the confirmed census later. Perhaps, they'll act like the ones who don't see it as one, non-contradicting, and non-incompatible account are the problem in some way.

Apologists like to bring up the gender of the women at the tomb as if their words would need to be able to be used as testimony in a Jewish court in order to be believed, but you don't need that burden or criteria to be believed in conversation, which the Gospel of John has everyone believing the women fine, no problems. But testimony to a Jewish court is a good analogy.

Apologists like to justify the differences and contradictions between the Canonical Gospel accounts as four witnesses giving testimony, which for some reason they start talking silly about car crashes while ignoring how it taxes at inerrancy and the ability of the SAME Holy Spirit to inspire all at the same time.

My knowledge may only be Mishnah Sanhedrin deep, but in a Jewish court in accordance with the Hebrew Bible if you don’t have two collaborating witnesses, you don't have testimony, and you can't establish guilt, whether beyond a reasonable doubt -- similar enough to a Jewish court -- or by a preponderance of the evidence.

With the Immaculate Conception and the Virgin Birth, there aren't two witnesses, there isn't testimony, and unlike other aspects of the Jesus' life, they don't establish or prove anything that can be supported, not even by the Jewish court metric.