Turns out, there are 2 perforated tabs on the back that run the length, top
to bottom. When installed, a massive amount of thick plaster is molded all
over the back of the soap dish and squooshed (yeh, squooshed) into the wall.
The plaster mushes out and sticks to the back side of the drywall. Some of
it had broken off over time, so the fixture was loose in the hole, but
enough was still in place to keep it from falling out. A good swift yank,
and it shattered, and the soap dish came out, good as new.
Sorry Art, no screw.