# Are there any non-gauge invariant physical theories that describe anything from the "real world"?

I have learnt QFT and the Standard Model and we always had gauge invariance there (in my lectures at least...) I wonder if there are any theories which are not gauge invariant that describe a bit of "reality"?

I have read this post Can we allowe gauge non-invariant terms in a gauge theory? and this one What is the problem with a Lagrangian that isn't gauge invariant? but I could not find a definite answer to my question.

I have also read that the superconducting phase transition breaks $$U(1)$$ gauge symmetry, but I don't fully understand that bit, it doesn't appear to me a superconducting phase is described by a non gauge invariant Lagrangian, or anything like that.