Bit of a sequel to a post I made a year ago for some of the Engage characters about transferring to Garreg Mach and what house they would choose; I figured I'd do the same for Cai and his friends. If they were to go to separate houses, Cai would go to Blue Lions, Tialla to Black Eagles, and Peter to Golden Deer.
Cai's mother is a goddess and would align him well with the Kingdom and Church and is a very skilled fighter. He would get along really well with Ashe, Mercedes, and Annette. I can also see Felix not being interested in him, but after seeing Cai launch a fireball out of his hand directly at Dimitri, he'll immediately call dibs on sparring with him.
Tialla would be all over Linhardt to have him teach her all about Fodlan's history and be book buddies, while Linhardt does everything he can to get her off his back. She would be best friends with Dorothea and even earn respect from Hubert due to her intelligence and reminding him of his younger sister.
Peter would get along with Claude, Ignatz, and especially Leonie due to their archery skills and the latter two being village commoners. I do think Lysithea would give him the Ignatz treatment of trying to make him more confident, but eventually they would get along. Peter would be intimidated by Raphael at first but would quickly become friends by talking about their families and younger siblings.
If all three of them were to go to a single house, though, I'd probably pick Golden Deer for the many reasons I listed for Peter.
Which houses do you think Cai, Tialla, and Peter would choose?