The violation of Bell's inequality is an indirect observation of the preferred foliation
To clarify the meaning of "indirect observation": Different from a direct observation, in an indirect observation we do not really "see" the object in question. Instead, we have an observation such that all realistic explanations require the presense of the object in question.
But with this meaning of "indirect observation" the ATM becomes a consequence of Bell's theorem. A realistic explanation of the observed correlations is something which fits into the assumptions of realism (including realistic causality) as used by Bell. If we add Einstein causality, we can prove Bell's inequality, which is violated. Thus, we cannot add it, so that all realistic explanations have to violate Einstein causality.
The existence of a preferred foliation follows from this.
Once according to quantum theory a violation of Bell's inequality is possible for arbitrary pairs of space-like separated events A, B. Applied to such a pair of events, there are only two possible realistic explanations: One with a causal influence A->B, and the other with a causal influence B->A. Assuming that there exists a relation of causality between events which does not contain closed causal loops, and which provides an explanation for all possible BI-violating experiments, this notion of causality has to contain or A->B, or B->A for every pair A,B of events. And, of course, this notion of causality has to be consistent with standard causality inside the light cones.
Now we can define the event contemporary to A on some world-line B(t) as the point t_0 where A->B(t) for t>t_0 switches to B(t)->A for t<t_0. If there is some t_- with B(t_-)->A and some t_+ with A->B(t_+), such a t_0 should exist.
Of course, we cannot detect this preferred foliation by observation. All the construction does is to prove it's existence, given realism and loop-free causality. So, for solipsists and other positivists this existence proof is not sufficient. Which is something I do not really care about.
One may certainly question realism. The violation of Bell's inequality defines a conflict between the two ingredients used to prove it - realism and relativistic causality. So one may ask why I think it is realism which should be preserved.
This question needs some methodological considerations. We do not have a conflict between theories, but between principles. We usually cannot test principles separately (which is sometimes possible for theories). So what should be the rules for judging which principle has to be preferred in case of a conflict?
I propose here some criteria for deciding between principles and look what they tell us:
1.) Restrictive power: The equivalent of empirical content for principles. While principles do not have own empirical content, they restrict the theories which follow these principles in more or less restrictive ways. The more restrictive, the better. A principle without restrictive power can be simply ignored, similar to a theory without any empirical predictions.
Application: Einstein causality for observable effects only (allowing hidden real causal influences) is not in conflict with realism. Realistic Einstein causality has no additional restrictive power if realism is abandoned. It reduces to observable Einstein causality. Thus, the combination of realism with observable Einstein causality is clearly more restrictive than a rejection of realism.
The same logic holds for observable and realistic versions of Lorentz symmetry.
The only principle which is in conflict with realism but survives this argument is manifest Lorentz covariance. This principle remains more restrictive than Lorentz covariance for observables even if we reject realism.
2.) Generality: In case of conflict, the more general principle should be preferred.
Application: Realism is clearly the more general principle. We can consider it even as part of the scientific method: The methodological principle that one has to search for realistic explanation of observable correlations is equivalent to realism in Bell's sense - realism simply defines the meaning of "realistic explanation".
3.) Conflict with other principles: The more general principle may be questioned too, but only in case if it is in independent conflict with other principles too.
Application: It is not realism which is in conflict with other principles, but relativistic symmetry. In particular, it is in conflict with absolute time and contemporaneity, the Hamiltonian formalism, local integrable energy and momentum conservation, the de Broglie-Bohm pilot wave interpretation as well as physical collapse interpretations of QT.
4.) Compatibility with existing theories: One could argue that a principle is not viable if there exists a domain where no theory compatible with the principle exists, while theories compatible with the other principle exist. (This is the criterion most biased in favor of the mainstream, because it needs resources to work out theories.)
Application: Given the de Broglie-Bohm pilot wave interpretations, quantum theory is compatible with realism. Relativistic gravity too Snip.. Given my construction Snip. of fermionic field theory from canonical quantization, one can extend the pilot wave interpretation even to fermionic fields (for bosonic fields it is standard).
Relativistic QFT is in fact not completely manifestly Lorentz covariant. At least the canonical formalism certainly isn't. While this is usually not relevant, it becomes relevant because of the conclusions of point (1) that the principle which competes with realism is only manifest relativistic symmetry. Snip.
Thus, even this criterion fails to support relativity. Thus, a reasonable comparison of realism and relativity using neutral reasonable criteria for comparison of physical principles clearly prefers realism.
And, once we accept realism, we have to interpret the violation of Bell's inequality as an indirect observation of a preferred foliation.