An important criterion for the comparison of conflicting physical principles is the existence of various theories compatible with each of the two principles in conflict. Especially important is, of course, the existence of corrobated theories in different domains, which are compatible with only one of the two competing principles.
In this case, one would prefer the principle which is compatible with these theories.
It is important, in this case, to use only theories which are really incompatible with one of the principles. Instead, it is completely irrelevant, if the theory has been developed by a proponent of one of the principles, or if it is usually presented in a way compatible with only one of the principles.
One has to note, that, in comparison with the other criteria, this criterion prefers the currently established principles in an inappropriate way. Indeed, alternative proposals have to be worked out. This needs time and human power. That there is no alternative version of a theory for some special application can have a very simple explanation: Nobody has tried yet to develop such a version. Or, if to construct such an alternative version is problematic, not enough time has been spend to solve it.
In our case, the more popular principle is relativity, thus, the criterion of compatible theories favours relativity. And, indeed, at a first look, there seems to be a clear advantage on the side of relativity: The most important theories of modern physics are relativistic theories. On one hand, the standard model of particle physics — a special-relativistic quantum field theory. On the other hand, we have general relativity.
Moreover, it is widely believed that quantum theory is incompatible with realism.
But this first impression has more to do with ignorance of realistic alternative theories than with problems of realism.
The claim that quantum theory is incompatible with realism is simply false. The pilot wave interpretation (also known as de Broglie - Bohm theory or Bohmian mechanics) is a clear counterexample. It is a realistic interpretation of quantum theory, which makes predictions which are (in case of the so-called quantum equilibrium) identical to quantum theory. This proves that there are no compatibility problems between quantum theory and realism.
There are also other realistic alternatives: Especially Nelson's stochastic interpretation is worth to be mentioned. It is also equivalent in it's predictions with quantum theory. Then, there are realistic collapse interpretations.
Last but not least, even the popular many worlds intepretation claims to be realistic. But, given it's problems with introducing a notion of probability into the interpretation, as well as it's claim to be Einstein-local, we are unable to verify that it is really a realistic interpretation.
Relativistic and field-theoretic generalizations of pilot wave theories are also well-known. Especially important is Bell's proposal
J.S. Bell, Phys. Rep. 137, 49 (1986), reprinted in J.S. Bell, “Speakable and unspeakable in quantum mechanics”, Cambridge University Press, Cambridge (1987)
An interesting property of Bell's proposal is that one does not need to construct hidden variables for all quantum particles. Especially Bell's proposal defines hidden variables only for fermions. This approach extremely simplifies to question of existence of realistic interpretations: We do not have to care about pecularities of gauge fields, Higgs fields, or gravitational fields — realistic beables for fermions are sufficient for a realistic version of the full quantum theory.
On the other hand, this approach is less beautiful in comparison with a maximal set of beables. Moreover, Bell's version is not deterministic, contrary to other pilot wave interpretations. Therefore, the search for better, more beautiful realistic interpretations is far from being finished. See, for example, quant-ph/0506243 for a discussion of various alternatives. But this search for better theories is irrelevant for the question of compatibility of realism with modern quantum field theories. This question is positively solved by Bell's proposal. It is disagreement about the simplest, most beautiful variant of a realistic interpretation.
If we do not reject realism, then the consequence of the violation of Bell's inequality is the existence of a preferred frame. This can be proven: We have to assume that for arbitrary pairs of events A, B, it is possible to violate Bell's inequality for measurements made in some arbitrary small environment of A and B. In a realistic theory, this allows two realistic explanations: A causal influence A → B, or a causal influence B → A. If one knows, for all pairs of events (A,B), the direction of this causal influence, and if these causal influences do not have causal loops, then this allows to define a preferred foliation. In the domain of applicability of relativity this preferred foliation remains unobservable. We can be sure, that one and only one of the explanations (A → B), (B → A) is true, but we cannot establish which of them is true. Nonetheless, it necessarily exists in a realistic interpretation.
Thus, realistic theories in the domain of quantum theory require a preferred frame. Now, for many physicists this is already sufficient to reject it. The more tolerant variant of this is to consider a theory with preferred frame as having a serious problem or fault. As a consequence, realistic interpretations in the domain of quantum theory have a bad reputation among mainstream physicists.
Should we take this bad reputation into account in our evaluation? Certainly not. This would be circular reasoning. The necessity of a preferred frame in a realistic theory is a consequence of the conflict between relativity and realism we consider here. The criterion of compatible theories makes no sense if we allow the rejection of theories incompatible with principle A, because they are incompatible with principle A. In this case, it would be simply impossible for principle B to win.
The only question is if there is independent evidence against a preferred frame, which could independently justify a rejection of theories with preferred frame. One could argue is that general relativity is incompatible with a preferred frame. But, first, GR is a relativistic theory, and it would be circular reasoning to require from a realistic theory compatibility with a relativistic theory. Then, global cosmology — the domain of application of GR — even gives us a natural candidate for a preferred frame — the frame of the background radiation. Last but not least, with the general Lorentz ether theory proposed by the author we have an alternative theory for the domain of relativistic gravity which is compatible with a preferred frame.
Thus, a bad reputation based on prejudices against a preferred frame should not count for the purpose of this discussion. Given the strong prejudices against a preferred frame in the mainstream, it seems impossible to distinguish bad reputation caused by the preferred frame from bad reputation caused by other, independent problems. Therefore, it seems necessary to ignore the reputation of a theory completely in the discussion of this criterion. The only relevant question is, therefore, if such theories exist or not. And this question has a clear and certain positive answer: they exist. To prove this, Bell's proposal is sufficient.
General relativity is a classical theory, thus, itself a realistic theory. Thus, there is no compatibility problem with realism.
One could argue that string theory has solved the problem of quantization of GR, thus, there is a relativistic theory of quantum gravity. But this relativistic theory described gravity as a spin 2 particle on a Minkowski background. For theories with Minkowski background it is not problematic at all to introduce a preferred frame. Thus, in the only known relativistic quantum theory of gravity, the introduction of a preferred frame is unproblematic.
While I don't know if there is some version of a pilot wave theory for string theory, I see no reason to expect other problems, and, given the current situation in string theory (where even the definition of the theory is not known) I see no reason to care about this highly speculative direction of research.
Up to now we have looked for realistic alternatives of existing relativistic theories, and have found them. Thus, despite the advantage the criterion of compatible theories gives to the principle preferred by the mainstream, we have shown that this criterion does not favour relativity.
But, maybe, it will even favour realism? If we introduce into the discussion the condensed matter interpretation of the SM proposed by the author of these pages, then we have such a realistic theory. It is a theory which is more fundamental than the standard model of particle physics (SM), and allows to explain the particle content of the SM, especially to compute the SM gauge group as a maximal group which fulfills a simple set of natural principles. As the relativistic competitor of this theory, we can consider string theory. But string theory has not been able to predict a single property of the SM, and there seems no hope to compute the particle content or the SM gauge group with string theory.
Some properties of the model suggest that no four-dimensional spacetime theory will be able to repeat the results of this model. Indeed, the three generations of fermions, the three colors of strong interactions, and the three generators of weak interactions are associated in this model with directions in our three-dimensional space.
Now, this theory of the author is not yet established, but it has been already published in a peer-reviewed journal:
I. Schmelzer, A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Foundations of Physics, vol. 39, 1, p. 73 (2009)
If we include the theory proposed by the author of these pages — the cellular lattice model — into the consideration, the criterion of compatible theories gives an advantage for realism.
If one doesn't, based, say, on missed acceptance of this theory by the mainstream, one would obtain an even count.
Given that the criterion favours the mainstream preferences, thus, in our case, relativity, this seems a very poor result for relativity.