...water spots, as water spots should be soluable in water. Could be spots from some kind of solvent, or blood or poop. If you tried everything else, and want to try an abrasive cleaner - try toothpaste. Very mild, and I have used it on displays before without damaging the screen.
I had the same issue after hitting some very rough water and all the stuff that goes on with fishing and they did not just come off wiping the screen down so I brought mine in the house and I took dawn dish washing soap and some water and scrubed the screen with a terry towel and the spots came off. Do it lightly as not to crack or bend the screen but use some elbow grease too. You will have to find the medium. They will come off.
That anti glare stuff never lasts on any of my Furuno units. I just wait till it gets marked up a bit then buff it right off. Clean glass after that. Much easier to deal with. Don't notice much glare difference. I just use a mild cleaner type spray wax to get the anti glare stuff off. It does take a lot of rubbing but it does come off.
Bad spots on my Furuno screen that water, mild soap or glass cleaner would not remove. Got some Novus System plastics cleaner from one of the electronics web sites and it took it right off. There is a medium and a fine cleaner plus a pre wash that comes all together. Don't buy the large size like I did because its a lifetime supply. Regards, Jerry.