The proof against local realism is humankind’s the most relevant finding in the realm of science and philosophy; there is no excuse to be ignorant about it while pretending how interested and knowledgeable about awesome cool science one is, multiverses and all. OK, it was a difficult topic once maybe, but from now on, there is no excuse at all anymore:
Any good high school student, certainly any university undergraduate is able to follow the arguments step by step completely as long as there is no religious/obsessive hurdle to overcome internally.
Here the promised study guide (see below) to the disproof of local realism and the solution of the Einstein Podolsky Rosen paradox via modal realism. The complete work is available online here on Science2.0 and as a preprint, which is written as pedagogically as a scientific article addressed to experts allows, so I suggest newcomers to try read that article first and then switch to the Science2.0 posts, which have been repeatedly revised to be as didactic as possible, wherever the article should be too abstract.
The following listing of posts should help to read the preprint article which is structured almost like the posts; there is a post for almost every section you might have difficulties with. Start with the article’s introduction and then appendix (!) or straight with the appendix without ever bothering with the introduction (!) and switch to the posts immediately upon encountering difficulties:
1) Disproving Local Realism and Hidden Variable Madness are summarized in the preprint’s appendix. The experimental setup is introduced and the important ground set: Quantum physics disproves local realistic models. It may be helpful to also read about the Quantum Randi Challenge, which will make it yet clearer that absolutely no locally realistic model can be quantum mechanical. However, the Quantum Randi Challenge is a side issue here.
2) Spooky World or Crazy Mind: Nonlocality versus Antirealism is somewhat of an introduction that should help to understand that there are two sides to local realism, namely (Einstein) locality and (direct) realism. This post may help to understand the introduction and thus the general problematic, but it is not necessary to understand the main arguments.
3) The classical model is introduced in Many Worlds by Splitting a Wiener Sausage. That model is simplified further in Many World Sausage with Correct Quantum Factors. The latter article shows that the model is able to come up with desired factors needed in a quantum model. If you do not understand these posts fully, it should be OK, as long as you grasp that these models have part of the desired non-locality arising from the many worlds structure.
4) Classical Parallel Worlds and Local Non-Locality yet again clarifies that although the correct factors may be found in the initial sausage model, the model is still classical. The important direction "DirectlyReal" is introduced.
5) Gambling in the Multiverse: Empirical Probability versus Classical Fair Meta-Randomness explains the difference between classical probability and quantum probability and why the probability that we empirically see in experiments is the quantum probability. This post contains a tree that was too off-topic for the preprint article yet it should be very helpful to most readers!
6) Einstein Podolsky Rosen Paradox Resolved by Local Modal Realism finally explains how the sausage model is turned into a quantum model. It introduces the growing branches instead of cutting the sausage and discusses the crucial step that turns the direct realism into a modal realism.
7) Why Einstein Could Not Solve EPR Though He Could Have explains a little differently what is mostly discussed in the conclusions of the preprint article, namely that modal realism is a necessary paradigm change that was already indicated by plain old special relativity.
If you get stuck, please comment on where and why so that it can be improved. Tell us constructively where you get stuck although you seriously tried (more than five minutes, like, sleeping over it).
(I will ensure the comment thread here is going to be helpful for readers, so pseudo-scientists and religious anti-multiversers please stay away or get deleted.)
Again, start with the preprint: S. Vongehr: "Many Worlds Model resolving the Einstein Podolsky Rosen paradox via a Direct Realism to Modal Realism Transition that preserves Einstein Locality." arXiv:1108.1674v1 [quant-ph] (2011).