How would a realist interpretation of the Mermin-Peres square look like? How would a realist interpretation of the Mermin-Peres square with counterfactual definiteness and the existence of states prior to measurements look like?
 A: It's possible to save realism.
It's counterfactual definiteness which has to be sacrificed so that realism may live!!!
Retrocausality is one such way. The square knows in advance which row or column will be selected, and it instantiates real states only for those observables in that row/column.
Superdeterminism is another way. The experimenter may think he has free will in selecting which row or column, but he's deluded. The choice is FATED and PREDESTINED, and the square can divine the fate.
It's the rules of causality which have to be broken to save realism.
So you see, real definite states are still possible prior to measurements.  
A: The nobel laureate Gell-Mann and the renowned cosmologist Hartle came up with a "sense"ible "real"ist interpretation which doesn't "break" causality at http://arxiv.org/abs/1106.0767 .
There's no need to "break" causality by retrocausality or superdeterminism. No, no! Negative probabilities are enough. So, quantum mechanics is a "real"ist theory where probabilities may be negative. There is a "real" "actual" state out there. It's just that the probability for this "real" state may sometimes be negative. However, the authors are not clear on whether "real" implies real or not. :-0 How can we get away with negative probabilities? Well, probabilities are nothing but betting strategies! The whole world is a casino and we're all gamblers! Any fine grained choice of histories which admits negative probabilities is an inadmissible gambling game. So, the "actual" state may have negative probabilities, but we can never perform fine grained enough measurements to determine that. Good for us gamblers. It prevents us from losing money in a guaranteed manner in the grand casino.
A: Many worlds is realist.
In MWI, a Platonic realm really exists, and there's actually a Hilbert space in it, and there's an actual physical state vector in it. It describes the actual state of the square. Too bad there's no preferred basis for splitting for the square prior to measurement. Then, measurement entangles it with everything else, and so, the actual physical state vector can only be the universal wave function of the entire universe.
No preferred basis. But give me a basis, and there is an objective complex value for its component along a chosen basis vector. If you want it up to 8 significant digits, it can be $(7.1947279 - 7.4624267 i)\times 10^{-72}$, and if you want more significant digits, those are also objectively real.
That's realism for you, nonlocal realism where everything in the universe are entangled with each other.
