You mention it looks like they are beneath the epoxy, but hopefully they're on top? What type of cleaner have you used on them so far? If it were my rod, I'd try wiping them with water (you've likely already tried that) and then move up from there. Next I'd try wiping on some vinegar and distilled water, and letting it stay wet with that for about 10 minutes, then washing. Next I'd try some polishing compound for cars. Of course try to start on an inconspicuous spot just in case it clouds the finish.

Do you have sprinklers that use well water?

I'd do some googling about removing hard water spots from car finish, as that's a more common and frequently addressed issue that likely has the same solution.

It looks like something started to crystallize, or moisture caused some kind of blushing on the epoxy. Can you determine which layer is giving you the problem? If it is the topmost, you could probably fine sand it and put a new coat down. You might want to test that in a small obscure area.

It certainly looks like a response to moisture. Perhaps if you let it dry in the sun for a while?

Is the effect restricted to a finish epoxy layer? The effect is not found on the rod blank finish-right?

I am glad you got it to clean up, but you must tell us what you think happened, and why your cleaning method worked. Did some little woodland denizen decide to mark his scent?

My guess is out side dust concentrated into dew drops that later dried leaving mineral spots. The Pledge would probably have taken care of it, although I was going to suggest a paste wax. Dish soap is always a place to start.

The rod was refinished over 25 years ago with Dale Clemens Crystal Coat A & B and thinned with a little Crystal Flo. These products have held up well over the years considering their age and abuse they have taken.