Here’s the theory:
Tyrion Lannister is NOT actually a Lannister
Instead, he’s the son of the Mad King Aerys II Targaryen and Tywin’s wife Joanna Lannister. That has a lot of implications, including the fact that it would make Tyrion Daenerys’s half-brother. We’ll get into some of these.
But first, let’s admit that yes, this theory seems a little crazy at first. Tyrion not a Lannister? Why would Tywin treat him as a legitimate son, particularly when Tywin already hates Tyrion so much? It’s like a built in excuse. Plus, some would argue that it makes for bad storytelling, that it undercuts the troubled father/son relationship between Tyrion and Tywin.
I once was in that camp. But no longer. Tyrion is not a Lannister. We’ve been given all the proof we should need to know it’s true, and it’s going to be a significant reveal in season 6.