I am pretty sure this is not related to local realism. I would naively imagine the situation as follows (not having read anything related in detail and not being a physicist). The forces between particles become very weak with growing separation but never become zero. Now imagine throwing two ping-pong balls into a pool making ripples on the surface similar to the force field or the probability of presence of the particles. There will be a interference pattern and you can probably find a valley where you can place a third ping-pong ball and it will sit there more or less stable. Now finally imagine every pair of ping-pong balls forming the valley for the third ping-pong ball. This may of course be incredible far off from what really happens - just making up images.
I like the ping pong balls over the surface of a pool, that's a neat analogy. However, I am also not a physicist, but my impression is that this is something closer to how a http://en.wikipedia.org/wiki/Lagrangian_point (in particular, the L4 or L5 lagrange point) works. In fact, I think the size difference thing they describe in the article is probably very similar to how gravity, and relative object positions work out based on their mass ratios. Kind of an equilateral force balance, with a self stable state that things will naturally gravitate towards if the background energy level is low enough. Course, then that makes me wonder if there's a 4-body version with a stable tetrahedron formation?
