special relativity - Is causality a formalised concept in physics?

I have never seen a “causality operator” in physics. When people invoke the informal concept of causality aren’t they really talking about consistency (perhaps in a temporal context)?

For example, if you allow material object velocities > c in SR you will be able to prove that at a definite space-time location the physical state of an object is undefined (for example, a light might be shown to be both on and off). This merely shows that SR is formally inconsistent if the v <= c boundary condition is violated, doesn’t it; despite there being a narrative saying FTL travel violates causality?

Note: this is a spinoff from the question: The transactional interpretation of quantum mechanics.