Don't Bell experiments rule out local non-realism too? Bell experiments rule out local realism (hidden variables). But it seems to me that it also rules out local non-realism (no hidden variables).
Local non-realism makes 2 claims;

*

*Two distant events can't affect each other faster than light.

*Any measurement event where the observable is in a superposition, will have a random (weighted by the probability distribution) outcome. This doesn't reveal a hidden pre-existing value. Rather, it creates it. It creates information. It is 'fundamentally' random.

But Bell experiments show that two entangled particles, far away from each other, measured on the same observable, give results that are 100% correlated.
If both measurement events are non-real, they are both 'fundamentally' random and so it's impossible that they will give correlated results. They will give random results.
Two non-causally-connected, no-hidden-variables, random events can't give the same result every time. It is an explicit contradiction of the definition of random.
Thus, Bell experiments also rule out local non-realism.
So why do people say that local non-realism is valid? How is it not instantly rejected by the experimental existence of Bell correlations?
Local realism and local non-realism both fail.
 A: Bell's tests rule out local hidden variables (assuming statistical independence if you will). The idea is the following, you have two entangled particles and a single detector for each. You want to find the conditional probability $\mathcal{P}(AB|xy)$ that you measure $A$ in the first detector with settings $x$, and measure $B$ in the second detector with settings $y$.
Now you start by (1) assuming quantum mechanics is incomplete, and need some extra hidden variables $\lambda$ to explain the probabilities, thus you have some distribution given by
$$\mathcal{P}(AB|xy)=\sum_\lambda \mathcal P(AB|xy\lambda)\mathcal{P}(\lambda|xy)$$
(2) assuming statistical freedom, we can safely suppose that  $\lambda$ is uncorrelated to the measuring devices, as we can put them as far as we want (this assumption when thrown away gives rise to superdeterminism), so we write $\mathcal P (\lambda|xy)=\mathcal P(\lambda)$
(3) assume separability, this is often said to be causality or determinism, is given by the idea that $\mathcal P (AB|xy\lambda)=\mathcal P(A|xy\lambda)\mathcal P (B|xy\lambda)$
(4) assume that there is no action at a distance (no contextuality), the measurements of each detector do not depend on the settings of the oposite detector: $\mathcal P(A|xy\lambda)= \mathcal P (A|x \lambda )$ and $\mathcal P(B|xy\lambda)=\mathcal P (B|y\lambda)$
Finally we have
$$\mathcal P(AB|xy)=\sum_\lambda \mathcal P (A|x\lambda)\mathcal P (B|y\lambda) \mathcal P (\lambda)$$
and with this object you can build a correlation function that has an extreme value. This extreme value is violated by quantum mechanics experimentally and can be predicted using Schrödinger's equation.
Now, forgetting about (2), locality is often thought as assumption (4) [disclaimer: the terminology is very messy and sometimes it can mean something else]. Where is realism in all of this? Realism or Counterfactual definiteness (the fact expected values are defined before measurement) as asked by EPR is not very clear here. John Bell preferred to use the term "local causality" instead. Sometimes realism is targeted at (3), it is claimed that (3) is not really an assumption and comes from probability theory, thus QM would purely violate locality! However, some would argue that (1) is indeed what was meant by realism. In the vision of people like Niels Bohr, quantum mechanics was complete, the fact that it uses instead probability amplitudes seems to avoid any need of hidden variables.
Many people still argue that if you assume locality the measurement problem is still an issue and still needs hidden variables to explain the results of Copenhague theory (which remains agnostic and just postulate collapse). However there are decoherence theories that claim that you just need to study decoherence to recover the results of measurement, fixing Copenhague and still you would not need hidden variables. Other theories like many worlds interpretation go further, you assume Schrödinger's equation is all there is, and the different tensor products in your states represent different worlds. In this sense these theories remain local and reproduce all the spookiness. In these theories there only exist the quantum state and that can be only modified locally according to Schrödinger equation.
A: 
Bell experiments rule out local realism (hidden variables). But it seems to me that it also rules out local non-realism (no hidden variables).
Local non-realism makes 2 claims;
Two distant events can't affect each other faster than light.
Any measurement event where the observable is in a superposition, will have a random (weighted by the probability distribution) outcome. This doesn't reveal a hidden pre-existing value. Rather, it creates it. It creates information. It is 'fundamentally' random.
But Bell experiments show that two entangled particles, far away from each other, measured on the same observable, give results that are 100% correlated.

Realism is the position that measurable quantities are described by hidden variables. Non-realism, the idea that realism is false, doesn't imply any particular position about what happens in Bell experiments.
There is a local theory that explains Bell correlations: it is called quantum theory.  The physical quantities that describe the evolution of a quantum system are Hermitian operators that evolve locally:
http://xxx.lanl.gov/abs/quant-ph/9906007
http://arxiv.org/abs/1109.6223
The correlations arise as a result of locally inaccessible quantum information being carried in decoherent systems, not as a result of non-local influences.
A: Claim 1 is that the theory is local. Claim 2 is that there is wavefunction collapse onto a randomly-selected eigenstate at the time of measurement. The long-distance correlations in outcomes seen in Bell experiments in the context of Claim 2 do indeed show that wavefunction collapse must be non-local. But Bell experiments are only excluding the possibility of local theories of wavefunction collapse, not Realism or non-Realism.
The Everett Interpretation (also known as the Many Worlds Interpretation) has no wavefunction collapse, and is both local and Realist. There are no faster than light influences. The wavefunction represents the true, objective state of the universe. It is also deterministic - there is no randomness. When an observation is made, the observer enters a superposition of observer states each seeing one outcome. This happens locally. In a Bell experiment, two particles with correlated wavefunctions are separated. Each observer observing a particle becomes correlated with it, and hence with the other particle, and hence with the other observer. When the observers get back together to compare notes, each superposed sub-state only interacts with superposed sub-states of the other observer making compatible observations. You get perfect correlation without faster-than-light influences, because there is no collapse and loss of information at the time of observation. The information is retained, but is not directly accessible to the observer sub-states who cannot see one another. They can only deduce their existence because interference between them cancels out certain combinations of outcomes.
An electron passing through one slit cannot 'see itself' passing through the other slit. (They are not, for example, electrostatically repelled from one another.) An electron (if it was smart enough!) could only deduce that it was part of a wavefunction when it observed that it never hits the screen at the nulls of the interference pattern.
Local realism can survive Bell's test. What gets contradicted is local wavefunction collapse.
On a philosophical note, some people argue that because the non-locality of wavefunction collapse has no observable consequences (it can't do, because there is a local alternative interpretation that predicts the same observations), and in particular, can't be used to send signals faster than light, that this collapse isn't really in violation of locality, but some more subtle property. Here, we have to point out the difference between ontology (our theory about what is) and epistemology (our theory about what we can observe/deduce). Wavefunction collapse is an ontological theory. We make an observation, and it collapses the entire wavefunction onto a single randomly-selected eigenstate instantaneously, everywhere. Observable consequences or not, this is the picture in our head.
That ontological picture has such strange consequences when combined with relativity that many reject the question entirely, and say there is no objective ontological 'reality' out there for us to observe. There is only epistemological observation. Quantum mechanics gives a method for calculating the outcome of experiments, but it should not be taken as saying anything about what is really happening behind the scenes. We observe shadows on the wall of Plato's cave, but nothing is outside the cave casting them. There are only the shadows. This is non-Realism.
As Einstein put the question: "Do you really believe that the moon isn’t there when nobody looks?"
That physics should take this proposal seriously - when there is a local, deterministic, realist alternative interpretation readily available - I think is fascinating in terms of the psychology and sociology of the scientific community. Einstein was wrong about hidden variables, but I think he was right that there was a problem.
Whether you choose to believe in non-local wavefunction collapse, local superpositions of observers, or that there is no reality to observe is your choice. They all make the same predictions about observations, so they are experimentally indistinguishable. None of them can ever be shown to be wrong. But the existence of the Everett interpretation as a local, realist ontology means that Bell experiments cannot exclude either locality or realism. They can only do so in combination with particular metaphysical assumptions.
