I think Yata is correct, and that the emperor's soul went to the great beyond. At the beginning of the story, the emperor is essentially a Lifeless. A soulless corpse, animated by investiture. The wonder of what Shai did was to create a new soul; one that matched the old soul enough to fool everyone, including the emperor's corpse. If it wasn't close enough, the corpse would have rejected it, and the forgery wouldn't have stuck.
If the soul was just disconnected or blocked, like Wrieth suggests, then Shai's task would have been different. She would have been asked to reattach or reconnect (find) the soul, not copy it.
As far as what happens when the emperor dies, I think that depends on how well the soul sticks. If he has to re-stamp himself until the day he dies, then I think the soul will just fade away like the stamps on Gaotana's arm. If the soul becomes "realmatic" enough to no longer require the stamp, then the forged soul could probably hold together enough to pass to the great beyond, where there will be two of them.
This would put Shai's accomplishment up on par with, if not greater than, the creation of Nightblood.