As you can see from the state of disrepair of the original plaster and lath ceiling, and the fact that somebody chunked a whole bunch of cheap wallboard up over it at sometime in the 1970s or 1980s, this is not the first time this issue has occurred.
Ebear the Headless Ceiling Repair Gunner.... (thecoughlin down below, trying not to inhale too many mouse turds...)
It turns out that I was right: the problem was the overflow drain, and in particular the fact that the gasket around it had petrified.
But hey, I got to tear down a ceiling today, and get a face full of mouse shit and old sawdust and bent nails (we dug out a pretty substantial mouse nest in the process). Other trophies recovered from the space under the tub include a petrified bottle of Nair, some cleaning fluids, and a certain number of (waterlogged) feminine hygiene products (in original wrapper and otherwise pristine condition).
And a piece of mouse artwork:
In conclusion: Nair if you dare wear eye protection!