I can’t search on here as I’m using a mobile reader, but didn’t we have a similar q a while back? I remember making the connection as I see similar brown spots on my bathroom door and walls. I think the consensus was iron or mineral spots from dried water drops?
I had initially thought some glop from cigarette smoke, but that would be on or near the ceiling, not clustered around areas that might possibly offer insect access.
Back when I managed apartments, we had a retired tenant who smoked pipes all day long and lived there for seven years before he passed away. We saw evidence of the oils from tobacco (a yellow/brown) everywhere. We wiped down the wood cabinets and came back a week later to find the oil had literally oozed out of the wood to create a new sticky (and smelly) coating. One of the many things we did in cleaning was to replace all the electrical plates because both front and back were just coated in that stuff. (A year later, after one set of tenants had moved out because of a pervasive tobacco smell, we still found this tobacco oil oozing out of wood all over the apartment.)
If there’s no odor that you’re finding, it could still be a residue of some other kind of aerosol - incense, hairspray, oil lamps - that has coated the backs of the plates and mirrors and is dripping out as humidity and temperature in the bathroom change.
We get that a lot if we leave the bathroom window open a crack. It’s basically fine grit and dirt from outside that gets caught up in the shower steam and condenses on the walls, etc.
This was the previous thread I was talking about, but it seems your brown spots are different, as the ones we were talking about were more like dried residue from water spots, not sticky stuff, and wipe off fairly easily.
Ok, this is pretty vulgar too, but do you have small children or really gross roommates who could be putting boogers on stuff? I totally did this as a kid and tried to pass it off as mildew when my parents complained.