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I think the best answer to this question is given by Roderich Tumulka in http://de.arxiv.org/abs/1501.04168 . Among many other papers on this subject, the Tumulka's paper stands out by clearly distinguishing 4 different notions of "reality", which he calls (R1), (R2), (R3) and (R4), and analyzing each of them separately. He concludes that only (R4) is the assumption of Bell's theorem. But he also points out that (R4) is the mildest form of realism, that it is very hard to abandon it, and that he (Tumulka) takes it for granted, just as Bell did. By taking (R4) for granted, he concludes that Bell's theorem does not assume reality.

I absolutely agree that (R4) is the only reality assumption in the Bell theorem. I also agree that it is very hard to abandon it and hence that it is quite natural to take it for granted. Nevertheless, I do not think that it is absolutely impossible to abandon it and absolutely necessary to take it for granted. Hence, for the sake of completeness, in my "Solipsistic hidden variables" paper http://de.arxiv.org/abs/1112.2034 I explore the logically consistent (even if philosophically unappealing) possibility of an interpretation in which (R4) is explicitly abandoned, to better understand in what sense nonlocality could be removed (or at least reduced) by an explicit rejection of reality.

What I would like to see a discussion about, is whether others agree that (R4) is really the only assumption of reality that is relevant to the Bell theorem and whether it is reasonable to question the validity of that assumption.