[Verginius Rufus]

I don't know if any long-term alliance between the Turks and the Byzantines/Rhomaioi/Greeks/Eastern Romans was possible given the religious difference. The Turks were Muslims and wanted to extend the areas under Muslim rule. The Ottoman state began around 1300 on the frontier between the Islamic world and the Christian world, and attracted zealots who wanted to fight to extend the land under Islamic rule.

How could the Byzantines make allies with people like that?

True, it would have been nicer if instead of fighting they had organized seminars to discuss Homer, Plato, Sophocles, and Vergil.

[SunkenCiv]

The Turks also knocked hell out of a bunch of other muzzie states and empires, building a good-sized caliphate in the process; and the Byz had that cultural legacy they sought. Lepanto was the last sea battle (large one anyway) to use ancient-style galleys, the ancient ways were still vital. And the Turks could have used an alliance with somebody, kinda like now, they piss off everyone they meet. :’)