My answer may be a little contentious, but my view is that while the mathematics in a theory must not have clear mistakes, mathematics does not make physics; it merely describes it. A mathematical equation is simply a statement in a somewhat condensed form, thus F = ma simply says the force on an object is given by its mass multiplied by the perceived acceleration. That applies always, although in fairness the statement relating to the Lagrangian in the standard model of particle physics would be somewhat cumbersome. However, the point I am trying to make is truth is not determined by whether the mathematics are consistent, but whether they are correct physically.
If a theory makes predictions, then it is falsifiable, although there is the caveat that theory may make a prediction of what happens, but concede that there are additional effects, i.e. the theory gives, say, an expectation result that should follow statistically if enough samples are taken, or alternatively, it might be unfalsifiable because it says this is what happens first but something else might follow or some overlooked effect might apply additionally. In each case, a result that does not fit might not falsify. Quantum mechanics gives a good example of the first. If I predict the position of one electron fired at two slits, the Uncertainty Principle and the diffraction effects will mean I cannot predict where that electron will be, but the physics are perfectly sound. For the second, I can have an equation that is correct for the trajectory of an arrow, but if you overlook the wind, an erroneous landing cannot falsify Newtonian dynamics.
Finally, there is another aspect. I have a theory relating to planetary formation and biogenesis, and I have made over seventy predictions. Some of them are conditional, because they are the "expectation" sort, but in general they all have problems of not being able to be applied because so far you cannot carry most of them out. For example, one of them predicts what you will find if you go and dig reasonably deeply at the bottom of Hellas Planitia on Mars. I regard that as a useful prediction that justifies what led to it being a theory. Obviously, it may not be correct, but that is irrelevant to whether it is a theory. One day it may be tested. However, if the theory is incapable of making any prediction that could be tested, or will not lead to something that could be tested I regard it as not a theory, but more fiction.