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alternatehistory.com
the challenge is at some point in its history the Ottomans begin to extend in Europe but the never monopolize control of Anatolia or get into the Middle East. And by 1600 the have lost their last Asian territory.
If we can achieve this tall order is it possible for this alternate Ottoman Empire to still survive into the 20th century?
2nd option - the Ottoman Empire expands as much as OTL or nearly so, but fragments into Asian and European halves. These two halves can have wars of attempted reunification with the European empire and the Asian briefly occupying part of each other, but never succeeding for long periods.
Even if neither formally accepts the situation the practical separation is pronounced enough that that historically they are remembered as separate empires.