I wouldn't think Sues have souls. The way the PPC sees them, they aren't people - which is why it's okay to kill (and occasionally eat) them. In the case of continua with a canonical afterlife, even a really easily accessible one like in MSPA, I don't think a Sue would go there just because she was never really a person in that world to begin with.

Now, I'm wondering: if an agent dies on a mission in such a continuum, what would the results of that be?

It is my own personal view that Mary Sues/Marty Stus do not have souls if only because chasing them down in the afterlife to kill them again is needlessly tasteless (a view which others have reflected in previous comments.) Plus - from a purely structural viewpoint - the assassination of the Sue should be the climax of the mission. Doing it twice only hinders the overall story.

It should be taken into account that some Sues or Stus might be created as ghosts or spirits. *glares at the Danny Phantom continuum* Even in such a situation, one wouldn't need to go into the Underworld to hunt down the Sue. Just grab an unlicensed nuclear accelerator and remember not to cross the streams.

The idea of a team assigned to pursue their enemies into the afterlife is a truly horrifying one.

...I'd read it. I'd read it, shake my head in horror and possibly shame, and write a scene in which the Flowers discussed the moral issues inherent in the department, but I'd read it.

Mary Sues, not being real characters, don't have souls.

That doesn't stop Agent Barid from shrinking their heads to trap their souls. He comes from, and works in, a world with frequent resurrections, so it is more of a habitual precaution than anything else.