...though I am a bit confused by how he describes it in the last line — doesn't that sound more like non-realism (or at least "shut up and calculate") than MWI?

Isn't the point of the "best" explanation (in the Bayesian sense) that it is the one most at peace with the "shut up and calculate" mentality? My reaction, which please feel free to disregard, is that nothing could be more "real" than saying something like, "Okay, here's the theory, it's self-evident given our observations. Great. Now shut up and multiply. Onto the next question."