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How could the Ottoman Empire, or a Turkish empire with similar extent, be Catholic, and what would the effects be?

As for how they could become Catholic, perhaps somehow the whole Latin Empire and Fourth Crusade somehow leads to some Turkish beyliks becoming Catholic, and they ally with the Latin Empire against Nicaea? Or perhaps during an Ottoman civil war a Catholic faction (how it arises, I don't know) takes power--but it would take quite some time for them to regain the Ottomans' territory, and they might not be able to regain all of it?
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