Don't know if this is relevant to your problem ..
I used to have 'black' (whites on neg - opposite to yours) spots on prints when I used tap water and Rodinal combination. Not always but quite often to be a PITA.
When I used the same tap water for other developers like D76 or XTol no spots whatsoever.
So, maybe there is something about Rodinal and 'impure' water.
Since then I use demineralised water to mix and dilute all my developers .. problem dissapeared.
Mid last week i reported this to Ilford Harman via their "contact us" page. Hope they get back to me - I offered to send them a scan or a clip from the film. My assumption is that if they physically inspect it they can give an explanation.
I use filtered municipal water for the rest of the process.
Still, my circumstamce may have nothing to do with the OP's problem.
@Michael W, The photo chemistry is designed with distilled water in mind - any other water is hit or miss.
Buy Yourself a distiller ~ $200, pour some water or filter some water and fill it up with it, wait for the distiller to complete, then check the thick gunk on the bottom of the distiller - this is the mineral, micro-bacteria etc. etc. content that gets into the emulsion when You process it without using distilled water.
Sounds like contamination in the water.
Are the spots raised off the surface of the film or entirely within the emulsion?
Modern chems are designed to be used with any water. I have a whole house filter on my system, and have never had problems with deposits on my film or paper. As insurance, add a faucet filter to your system or buy a filtering pitcher to clean your water prior to use. At different times of the year water companies flush their lines of sediment, occasionally some can get through to homes.