Does PhotoPrism write tags to metadata?

I am thinking about trying out PhotoPrism, but I want to know if the face tags are written to the file, i.e. can they be recognized by other applications? Can it read face tags written by other applications?

Not yet.

Including people information in our metadata export files is planned: People: Include people information in yml exports · Issue #1555 · photoprism/photoprism · GitHub

As well as reading face tags from other tools:

Longterm we also plan to write metadata changes back to exif/xmp (Write metadata back to originals · Discussion #1092 · photoprism/photoprism · GitHub)