How can you get a European* emperor on the throne of China? How can you get a European** Caliph? Or a fully black African Roman Emperor***?
They are separate scenarios (though all of them at the same time would be cool). They must be achieved before 1780 with a POD no earlier than 1 AD, and later POD, earlier occurrence is better.
*The European Chinese emperor doesn't have to be of a European ethnicity, but he has to be born in Europe if not.
**The Caliph does not have to be born in Europe but he must be of a European ethnicity.
***The black Roman Emperor only needs to rule half the empire, but no less than that. Can't be the ruler of a state in Egypt that claims to be the Roman Empire, unless that state also includes Anatolia and Byzantium or something.