To begin with, it is crucial to make a distinction between two different senses in which a theory might be said to be "relativistic". First, a theory might be empirically relativistic. This means that what it predicts for the outcomes of experiments will exhibit the usual relativistic properties — for example, it should predict the familiar relativistic behavior of clocks and meter sticks in relative motion. More generally, it should agree with classical relativistic mechanics about the behavior of macroscopic objects and it should predict (ignoring for the moment gravitation and general relativity) that an experimenter cannot tell whether an appropriately isolated laboratory has been set into uniform motion.
Despite the central role it is given in certain philosophies of science, however, observation is not everything. We thus need to recognize (at least) a second sense in which a theory might be said to be "relativistic" — namely, that it is compatible with relativity through and through, and not just at the (relatively superficial) level of empirical predictions. Such a theory will be said to be fundamentally relativistic. To make the distinction clear, it is helpful to contrast two different versions of classical electromagnetism. Let us call these the Lorentzian and the Einsteinian theories.
According to the Lorentzian theory, there is a physically meaningful notion of absolute rest (defined by the so-called "ether" rest frame) and a physically meaningful notion of absolute time. These correspond to the existence of a preferred family of coordinate systems over spacetime and the dynamics of the theory is defined, with respect to these coordinate systems, by the usual equations of electromagnetism. ...
By contrast, the Einsteinian version of classical electromagnetism is of course relativistic, not just empirically, but fundamentally. The notion of a really-existing but unobservable "ether" rest frame is dispensed with and all uniform states of motion are regarded as equivalent.
Sometimes it is thought (and taught) that certain experiments from the late 19th or early 20th century refuted the Lorentzian theory in favor of the Einsteinian one. But this is not correct. With regard to their empirical predictions, there is no difference between the Lorentzian and Einsteinian theories. ...
For our purposes, there are three important lessons here. The first is that the empirical violation of Bell-type inequalities does not require theories that fail to be empirically relativistic. But this is hardly sufficient to assuage the worry: if empirical relativity is the only kind of relativity that can be saved, it's not clear that relativity, in any substantial sense, is being saved. The second lesson is thus that, if we want to insist on preserving compatibility with relativity, it is fundamental relativity (not mere empirical relativity) that we must insist on.
But the third lesson is that perhaps abandoning fundamental relativity should be on the table as a serious option. Doing so would not necessitate empirical predictions at odds with the experimental results that are normally taken to support relativity. And it is clear that the use of a dynamically preferred but unobservable "ether" frame would make it very easy for theories to incorporate the non-local interactions that Bell's theorem (and the associated experiments) require.
Lorentz never said that there was a "physically meaningful notion of absolute rest" or a "physically meaningful notion of absolute time." After all, he achieved empirical relativity, so what would the physical meaning be?
This distinction between the Lorentzian and Einsteinian theories is an entirely modern invention, and a nonsensical one. In 1906, it was called the Lorentz-Einstein theory, and no one noticed any difference. There is some point in writing physics in covariant equations, but Poincare and Minkowski added that to special relativity, not Einstein.
There is no difference between a theory with a preferred frame, and one without one. Mathematically, a space is called a manifold if it has no preferred frame, but textbooks often define a manifold in terms of a particular frame. It is obvious that it makes no difference.
Some people, like Einstein, Bell, and the authors of the above essay have a strange belief that somehow quantum mechanics will make more sense if it more directly refers to unobservable hidden variables. It is live believing in an unobservable aether. Bell actually said that relativity makes more sense that way.
The above essay is wrong when it says that Bell's theorem requires nonlocal interactions. And it is crazier still to think that some trivial choice of frame for the aether will somehow make those interactions more comprehensible. If anything, Bell's theorem says that hidden variable cannot help understand certain quantum paradoxes. They are better explained by orthodox quantum mechanics.