Everyone is free to believe what they want to believe about these things. Until a time-travel machine is invented, no-one is going to be able to prove any of it one way or another. That's the account that has been passed on throughout the Christian East, going back to times when the ancestors of most Christians of Western European stock were still worshiping Thor and Odin and Loki and what not...
As to what the "clear" account of the New Testament is, well, perhaps its clear to you. But to the Church that exists where most of the Bible was written, and where they have been reading that Greek New Testament in Greek without interruption since the time the books were first written in those lands, the account I gave you is the one that has been held from the earliest centuries. They don't see a contradiction, and neither do I.
But I'm not here to try to convince anyone about whether it's true or not, just to state what the Orthodox Church tradition is.
Yes, we can prove these things because we have the Bible.
And God's Holy Word tells us that Mary and Joseph had sex.
And that is ok. :-) It's the way God intended things to be.