The Tarnished is the Soul of Cinder

Just to let you know, this is ASSUMING The Frenzied Flame ending is what starts the Dark Souls trilogy, with the Frenzied Flame withering after 1000 years and becoming The First Flame. The theory of each ending leading to another game in the Souls franchise is not canon. I just find it fun to speculate and see all the games as one huge narrative

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The Tarnished would technically be the first Lord of Cinder until Melina eventually kills them. Their spirit would remain due to being given Destined Death like Ranni, most likely staying inside The Erdtree as it slowly becomes more like the Kiln of the First Flame. The Soul of the Tarnished would slowly gain the power of every Lord of Cinder who sacrificed themselves to keep the Age of Fire going. Eventually, the Tarnished would take all of the power of The First Flame to manifest themselves in a physical form in Dark Souls 3 either as a last-ditch effort to stop The Unkindled Warrior from ending the Age of Fire or a final test to pass on the torch as the next Lord of Cinder, just like the Golden Order did in the form of The Elden Beast to the Tarnished.