AHC a christian Ilkhanate surving to modern day

The Ilkhanate was never Christian to begin with. Its rulers were mostly Buddhists who tended to favor religious minorities, particularly Christians, over Muslims. You might need to have the Golden Horde not convert to Islam.
 
Umm, very implausible, if not ASB. Religiously, the early Mongol dynasties tended to be pragmatists who favored whatever religion was politically expedient. In the Ilkhanate, this at first translated to allying with non-Muslim minorities (who depended on the state for their positions of power), and later, conversion to Islam to make their rule more acceptable to the population and wider political elite. For a Christian Ilkhanate to last, it has to somehow force Christianity on the populace at some point, which is very much a departure from the usual Mongol way of doing things.
 
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