Rabban Bar Sauma was a Turkic/Mongol Nestorian Christian monk who was sent as Mongol ambassador to Europe to meet with European monarchs, as well as the Pope, in attempts to arrange a Franco-Mongol alliance. The mission was unsuccessful.
If it had been successful, the Mongol Empire remained stable and ties between Europe, the Church and the Mongols have been strengthened what sort of flag would the Vatican be flying after a few decades or centuries?
This is my guess based on this version of alternate history.
I took the Catholic Flag (the crown and keys version is specifically for the Vatican, but often used interchangeable)
Add to that the top part of the soyombo from the Mongolian flag as that element appears in earlier Mongol banners circa the 1300s (or current Inner Mongolia separatists)
Replace the sun in the soyombo with the solar alisee pattee cross favored by the Nestorians and, viola/keman.
I was kind of torn between that color scheme and incorporating blue, as it is simultaneously Mongol, Uyghur and Marian, so I went with it.