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Craftworld Eldar use spirit stones to store their essense after death and return it to the infinity circuit of their Craftworld to fuel it and store their consiousness there forever while Exodites do much the same with their World Spirits.
But what of the Dark Eldar? They can not just leave them floating around: the Eldar background clearly states that merely left to its own devices an Eldar soul is drawn into the madness of the Warp and tortured there by daemons for all of eternity. Surely not a welcoming prospect or one that even the most extreme of mascocists would merely await?
What I think, is once a Dark Eldar would die, his/her soul would be consumed by Slaanesh. Infact, Slaanesh is sucking their souls slowly even when they live, this is why they suffer from the thirst. They have to feed on other souls to sustain themselves, or else they would become eaten by Slaanesh. So when they do die, I think that they do get sucked up and eternally tortured by Slaanesh.
The dark eldar feed on each others souls after death, its what keeps them from aging. Without this power to restore their well being, they would be no diffirent from humans.