The Khanate of the Golden Horde was the Longest Lived of the Khanates that succeeded the Great Mongol Empire of Genghis Khan and his direct successors. Ultimate, it collapsed during the 15th century (it was formally conquered by the Astrakhan Khanate in 1502).
How can the Golden Horde survive, and modernize to become a state in its own right, either as a major Islamic power or even as a Eastern Orthodox state?
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scenario #1
as an Eastern Orthodox state
The Golden Horde converts to Eastern Orthodox Christianity instead of Islam. The cultural, religious and power centres slowly moves closer to North-Western Rus (Russia, Moskovia). In the end by the XVIII century we have ATL Russia which has the proud name of "the Golden Horde" but it closely resembles OTL Russia with a few differences:
1) 80% of the nobility has the Mongol (Tartar) origin as opposed to OTL 50%
2) much more borrowings in this ATL Russian language from the Turkic language(s); Nobility might stay billingual Russian-Turkic and Turkic language is the second language of large part of Russian population
3) the monarch of the ATL Golden Horde is called "Khan" and at the same time "Tsar"
These Golden Horde Tsars/Khans will inevitably support Constantinople or/and want to take it as the centre of Orthodox Christianity. Which makes certain the clash with ATL analogue of Osman Turk. With unpredictable results.
And this ATL Golden Horde will have much more troubles in Central Asia against the Muslim Chagatai Ulus. ATL analogue of Tamerlane (if there is such) will concentrate his effort in fighting infidel Golden Horde which means even more trouble than in OTL. Actually that might mean total destruction of this ATL Golden Horde or the destruction of the steppe Turkic parts of the Golden Horde and the better preservation of the forest Russian parts of the ATL Golden Horde. And thus the Golden Horde becomes even more "Russian" as a result.
So all in all nowadays ATL we have the Golden Horde which very much resembles OTL Russia.
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scenario #2
as a major Islamic Power
The Golden Horde survives the invasion of Tamerlane, or Tamerlane dies earlier and there is no invasion.
And then this ATL Golden Horde changes its' policy towards Russia. Something like what the Osman Turks did in Orthodox Balkans - the same is done in Russia by the Golden Horde Turks. Meaning direct occupation, assimilation and forced islamization.
That might be even easier for the Golden Horde Turks in Russia than for the Osman Turks in the Balkans - the Catholic Christian neighbors of Russia will support the Golden Horde in its' effort to subjugate it totally. In OTL the Mongols often allied with Lithuania/Poland against Russia.
This ATL Golden Horde might ally with the Osman Porta against Persian Shiite Safavids and against the Christian West.
And internally the Golden Horde might resemble the Osman Turk Empire with janissares recruited from Christian Russian boys and the like. And I am afraid the Golden Horde will have the same violent disintegration as the Osman Turkey had in OTL.
But as Turkey of nowadays is the rightfull successor of Osman Empire there might have been some successor of this ATL Golden Horde in the 21-st century somewhere on Volga and Caspian Sea, maybe including the Crimea.