The existence of things like a negative square root in an equation have predicted the existence of things like anti-particles, and those particles have been found experimentally.
That's only word playing.
There's no more 'existence' in a negative square root, than to a positive one. You have to define what 'existence' means, and only then we can decide if there's some relation between anti-particles and negative square roots.
It's a false dicothomy to talk about math and 'physics' as separate things.