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.