the most likely sceneario would be for a pre 1260 where berke and nogai convert to chrsitnianity instead of islam , assuming that all the anatolian byzantine empire is conquered , so the european part is safe a new khante would be formed split from the ilkhante called , the purple khante or roman khante,
asumming since berke convertion to chhirstniaty prevents the halagu berke wars , the both can unite agianst the mameluks , most like conquering egypt , if that is the case a civil war will occur with halagu and berke on who gets who , in our timeline berke won.
assuming he does, the division would be , anatolia , armenia , the levant , syria and egypt for berke , mesopotemia and the rest of modern day iraq and iran.
(keep in mind that berke aslo has dominion over the golden horde so from his new base in constaninople an invasion of the balkans becomes way more possible)
once berke dies the roman khante and the golden horde would split but still be alies against the ilkhante.
going on to the 13th century , do to geography , history and one being islamic and the other chirstian the ilkhante and the roman khante
will become the sassanid persia vs Eastern roman empire. as figthing would ocurr,do to distrust of the muslims beacuse of the ilkhante
the roman khante would force covert the muslims , in some areas this would be easier espcially
the roman khante would be hit hard by the black death , and most likely suffer a lot i dont see it surviving past 1400 s to be honest, if internal striffe doesnt kill it timur would.
but it would have a lot of consquences for one , after nearly a century or a century and a half of chirstian dominion we would have a more crhistian populations.
the 8th and 9th crusades are butterflied away,
once timur is gone , antolia would fall in to many pety kingdoms .
assuming that the roman khante took control of many parts of the balkans , a new roman dynasty can take the mantle , and after timur is gone , and attack anatolia .
