https://en.wikipedia.org/wiki/Bell%27s_theorem says:

> In the vernacular of Einstein: locality meant no instantaneous ("spooky") action at a distance; realism meant the moon is there even when not being observed.

That's not what you said. MWI is a realist model; in fact, there's an argument that it's realism turned up to 11.

In MWI, there isn't the same moon there or even any moon in many "worlds"; its existence depends on your point of view's world. Realism in this context is about the idea that probabilistic events and measurements end up having a single objective result. In MWI, different points of view can see the same event have different results (because they're branching apart). MWI can be described as realist if you're considering all the branches real, but then you're not using the same definition of realism as the specific thing that the Bell Paradox is said to rule out.

By this argument, 3D perspective is non-realist, because a cube looks like a square from some perspectives and a hexagon from others – it depends on your point of view!

MWI is the most realist theory, because it says that everything is there when not being observed – in fact, it eliminates the concept of observation entirely. I think the tricky thing here is that MWI says that you are part of the system, which means that the intuition you've built up around collapse theories (where the “observer” has a privileged position outside and above the bits of physics they're “observing”) kind of falls over.

Wikipedia thinks it's realist.