Tema: Go East, Young Knight: The True Story of the First Crusade.

The unspoken fact is that, at best, the Byzantines were troublesome allies at the best of times, i.e. such as the oath of fealty that Emperor Alexios demanded from the Latin crusader-lords, neverminding that many of them had already liege-lords of their own or the harassment of the Crusader forces as they travelled to Constantinople by Pechenegs in the service the Byzantine throne. The Crusaders refused to hand over to the Byzantines largely because, by the time Jerusalem was finally taken, they'd come to regard the Byzantines as largely unreliable.

The bulk of the successes of the Crusaders have little to do with the cooperation of the Byzantines and more to do with the Crusaders themselves, often working hand-in-hand with Christians in the Holy Land and even apostate Muslims, sometimes of the fearsome sect of the Assassins (I've read a couple of anecdotes of the time that indicate the Assassins were often employed to remove a troublesome crusader-lord; it seems that the Old Man of the Mountain had little trouble in working with/for Christians).