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I think based on the article it's because of your sweat mingling with the chlorine.

Me: I'm allergic to cats.

Cat Owner: It's actually the cat saliva you're allergic to.

Me: facepalm

Same experience but in a hotel swimming pool with continuous recycling. After sitting next to the pool for an hour or more, with no one in the pool, I would dive in and find the chlorine too high and end up with eyes red.

Probably peed in by wildlife

Ambient pee if you will. Or background pee from the hose used to fill the pool.