Source: https://en.wikipedia.org/wiki/Greek_Muslims
Option1
Russian Empire creates a Trebizondian puppet state with an Ottoman prince as a figure-head.
Option2
Break-away state in Epirus/Ioannina headed by an Ottoman usurper that fails to take the Capital, ends up a puppet/protectorate of Italy
Option3
Osman (can't really have an Ottoman dynasty without its founder) never manage to expand against their Turkish competitors, and generally stay as a Byzantine vassal. They adopt the Greek language whilst maintaining their Muslim faith, and in a moment of opportunity conquer Byzantine lands south of the Marmara. Anatolia remains divided and the Ottomans remain a minor state at the edge of Europe, their Greek-ness solidifying. Sometime in the 16th-17th century, they conquer Constantinople and proclaim themselves an Empire, though never really expanding much after that.
Option1
Russian Empire creates a Trebizondian puppet state with an Ottoman prince as a figure-head.
Option2
Break-away state in Epirus/Ioannina headed by an Ottoman usurper that fails to take the Capital, ends up a puppet/protectorate of Italy
Option3
Osman (can't really have an Ottoman dynasty without its founder) never manage to expand against their Turkish competitors, and generally stay as a Byzantine vassal. They adopt the Greek language whilst maintaining their Muslim faith, and in a moment of opportunity conquer Byzantine lands south of the Marmara. Anatolia remains divided and the Ottomans remain a minor state at the edge of Europe, their Greek-ness solidifying. Sometime in the 16th-17th century, they conquer Constantinople and proclaim themselves an Empire, though never really expanding much after that.