The violation of Bell's inequality shows that one of the assumptions used in the proof should be false. This situation is quite typical in science: We often have situations, where different principles or theories have to be used to derive some prediction. In this case, the empirical falsification of the prediction does not tell us which of the principles has to be given up. All we know with certainty is that it is at least one of the principles used in the derivation.
One possibility is not to make any decision — to wait, until other observations give us more information. But even many different informations do not completely solve the problem: At some point, we have to make some decisions (say, to decide about the direction of the own future research), and the available information is usually not sufficient to derive a unique decision. Therefore, we have to use some heuristics.
One plausible heuristic is the following criterion of generality:
Criterion of generality 0: If, in case of a conflict between two principles A and B, principle A is more general than principle B, then it is reasonable to prefer principle A.
This is, clearly, only a weak, heuristic rule. If it would be a strong rule, we could never reject sufficiently general principles. Indeed, every experiment or observation relies on a lot of different principles and assumptions, and some of them are less general (starting with the assumption that the experimenters do not cheat).
The relativistic symmetry principle is based on a special symmetry group. This symmetry group is relevant only in a particular part of physics — relativistic physics. The symmetry groups of classical, Newtonian physics was different. As well, the symmetry group of Schrödinger theory is different. There is even a difference between the two variants of relativity theories — special and general relativity: in the first case, the symmetry group is global, in the second case, it is local.
In general, we observe that symmetry groups of physical theories often change. Especially they often change if we make approximations and limiting procedures. Symmetry principles can be, therefore, considered as principles related to some class of theories, which are usually unified by the same level of approximation.
For realism, the situation is completely different. Realism is a very general principle. Especially, if we interpret realism as a method (say, the methodological base for the search of realistic explanations), realism is applicable not only in physics, but in science in general, including humanities.
The greater generality can be seen already in our definition of realism. Especially, nor the definition of realism, nor the related definition of realistic causality contains any reference to special models of space, time, or spacetime. All what appears in the definition are abstract sets, probability distributions on these sets, and functions on these sets connecting them.
Even the proof of Bell's inequality can be done without any reference to a special model of space and time. Especially, we can consider a game in two different rooms, with the condition that there is no causal information transfer between these rooms, and prove Bell's inequalities for this game without any reference to light cones and other parts of relativistic theory.
The question which of the principles is the more general one is, therefore, easy to decide: The more general principle is realism.