This is not caused by ammonia, nitrite, nitrate, pH, GH or mixing a bit of distilled water with well water.
Distilled water is pure water that has nothing in it. It does not contain chlorine, chloramine, ammonia, nitrite, or anything. It is pure water and should have a pH of 7.0 and a GH of 0.
Reverse osmosis water is nearly pure water and can vary depending on the type of reverse osmosis unit. Good units will remove just about everything from the tap water, but cheaper units might not remove everything. As the units get older, the filters get blocked up and remove less too.
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The fish are breathing extremely fast and look fat. If they are normally that fat it is not a problem. However, if they got fat overnight and started breathing heavily at the same time, they have internal organ failure, which is causing the fat belly, and the fish is in pain and breathing heavily.
However, as you mentioned, tiger barbs are greedy feeders and will eat and end up fat. So the fish could just be fat and this has no bearing on the issue.
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I would say it is poisoning from something that has gotten into the tank.
Did you do a water change or add anything around the time this started?
Do you use a bucket specifically for the fish or any bucket?
Have you had the well water checked for anything and everything?
Well water should be checked by a professional water testing company at least once a year and preferably a couple of times a year. It can contain chemical runoff from industrial areas or agriculture and this can poison fish.
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I would add some carbon to the filter and look at filtering the well water with carbon before using it in the tank. I would also get the well water tested to make sure it's safe for you and your pets.
Make sure nobody is using any sprays or aerosols in the tank.
Make sure you don't have any creams, soaps, perfumes or anything else on your skin when feeding the fish or working in the tank.
Don't use anti-bacterial/ antiseptic soaps before working in the tank because they can leave a residue on the skin ad poison the fish.