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.