If Muslim then how come they weren't exterminated for leaving Islam? In Islam if you convert to an another religion you will get killed according to the Islamic law. How come these people weren't just exterminated? And how did those Turkmens convert to Christianity? Were there Byzantine missionaries in Seljuk Anatolia?
edit: Meant to say "Were Karamanlides' Turk ancestors ex-muslim or ex-pagan?. Sorry I forgot to add.
There are two main theories of their origins, but both predate Muslim conquest to the Christian Byzantine Empire.