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From the EPR article:

"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."

I do not see how a local quantum theory violates this notion of realism. Would it not be more accurate to say that the option is between non-separable realism and separable non-realism?

Norbert Schuch
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Realism in this sense states that measurable things have a value, whether or not they are actually measured. This lends a sense of objective realism to the measured quantities in the sense that they don't need a subject observing them to have that value. It's trivial to build a patchwork quilt theory which gets the answer "right" for everything observed, but lacks a value for things which don't have a subject actively observing them.

The issues that arise with Bell's theorem appear when you try to define these values without creating contradictions.

Cort Ammon
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Realism means that there are hidden variables that can explain the statistical results in a deterministic way. A non realist theory has no hidden variables, and thus the value of the variable to be measured could not have a defined value until it is measured. Here the measurement result is randomly chosen (within some distribution) at the moment of the measurement.

Notice that non-realism, with its inherent randomness, should not have any advantage in being able to reproduce quantum correlations. What magical process would enable this? Without non-local information a non-real local algorithm should be as bad as a real local one in reproducing quantum correlations.

I have asked multiple times to colleagues about this issue, and also read quite a bit, and I could never find an argument about why non realism could get away with non-locality. In my opinion Bell's theorem really show that QM is non local, because local non-realism is becoming just a word to avoid saying "no-local".

PS: some give QM as an example of a local non-realist theory, but this argument is circular, because it explicitly assumes in advance that QM is local.