How can I prevent files emptied out of the Trash from accumulating in "purgeable space"?
Based on this Apple support article, I gathered that the purgeable space might be taken up by local snapshots.  I had already backed up since emptying the Trash, but I have more than one backup drive.  I removed the backup disks that aren't accessible at the moment and some time later my system drive's purgeable space was shown to have been alchemically transmuted into free space.

Since Apple hasn't bothered to include information on how local snapshotting works when you have multiple backup drives, and since I never managed to discover a way to identify what is currently contained in purgeable space, I can't be entirely certain that this was what fixed the issue.  It would be quite helpful if there were to be a post listing the various ways that purgeable space can come about and how to prevent this from happening.  I haven't managed to find such a resource; in fact, the support article doesn't even include the word "purgeable"; perhaps comprehending such an arcane detail is considered to be beyond the mental capacity of Apple's target demographic, even for its support pages.