I think of this problem as a computer scientist. The universe is a giant state machine. Time is what separates one state from another. The laws of physics are a compact description of the state transition table. When we talk about "consistency", we are really asking: "How small is the transition table?" Because the most general answer is: any state can transition to any other state, stochastically. Such a universe, on average, would be utterly incomprehensible and would just look like noise if we could observe it from without. On the other end, the smallest transition table is the identity function: every state transitions only to itself. Such a universe is static, frozen solid for all time, no matter which configuration it starts in.
Many scientists would say that the "frozen" universe is extremely consistent, but boring. And they would say that the "maximally free" universe is chaotic and incomprehensible. But would they say it is inconsistent? I suspect not. But that is simply because this level of freedom hardly exists anywhere in our universe. If the freedom were dialed down to our universe + a few notable and observable exceptions, then there would be strong talk of "consistency".
For instance, if we found that the mass of the electron varied depending on which research group and machine was measuring it, but all our large-scale observations depend on it having a static, universal value, then some folks might tend to declare an inconsistency in the laws of physics. However, it should be clear that other folks would simply say there is deeper physics afoot and we just need to discover it (c.f. muon mass).
"Consistency" is basically presumed by scientists, and science is fundamentally built upon this presumption as an unstated axiom. If we admitted that physical law need not be consistent, then there would be effectively no way to discriminate between one theory and another. Take the Hubble constant, for instance. We have multiple values estimated by multiple teams. Astronomers generally believe that the Hubble constant has one value, and that there are deep, underlying reasons that different teams get different values. But in a world where inconsistency is accepted as possible and legitimate, there would be no reason to try to resolve the contradiction. The Hubble constant may take on different values depending on how it is measured, even if each of those methods were arbitrarily precise. Obviously, this kind of relaxation would make a hash of science.
Going back to the state machine, science is predicated on the notion that the "state machine" for the universe is indeed small. Small enough to write on a single sheet of paper at a size that the average human could read it. That's a pretty bold and remarkable idea, if we think about it! Instead of believing that the Standard Model is a fairly accurate description of the universe, we could instead accept thousands of competing, mutually incompatible models which each explain a small set of observations extremely well, while failing miserably on others (e.g., MOND). We would explain away the failures as "natural inconsistencies". After all, how could you possibly tell the natural ones from the unnatural ones?
So in this sense, science must assume that the universe operates according to consistent rules, because you simply cannot have objective science without this assumption. If you allow inconsistency, then every scientist can have their own theory, and you have no strong basis on which to object to it. Even a theory which predicts no observations could be correct, because it might explain something which will happen in the future but which has eluded our notice up until now.
Non-science has no such compulsion to believe in consistency, and indeed, religion practically demands that certain "inconsistent" events occur (i.e. "miracles"). At the same time, human nature (and indeed, all intelligence) must presume some level of consistency, because we need it just to operate in a hostile environment (predict food sources, threats, mating opportunities, etc.).