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How would it be possible to have one man to be widely recognized as being both a Mongol Khan, of the Borjigin line, as well as an Islamic Caliph, related to the Prophet Muhammad? This man would be directly descended from both Genghis Khan and the Prophet Muhammad. This person can also be descended from Genghis Khan's brothers or from Muhammad's more distant relatives, as was the case with the Umayyads and Abbasids. In order to count, this man needs to rule over a somewhat large domain. To give an idea of what constitutes "large", the khanate/caliphate should include four out of these five Islamic world cities: Damascus, Baghdad, Samarkand, Constantinople, Cairo. Barring that, it could encompass a large chunk of either India or China, holding major cities like Delhi or Beijing.

Ideally, this man would be a historical figure, but if you can create a plausible Great Man, that also works. This person doesn't need to conquer the whole area by himself, and there can be a Timur-style ruler behind the throne. The main goal is the state: how can an empire combining the Mongol Khanate and Islamic Caliphate be merged under the authority of one man in a plausible manner? Or would it be impossible?
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