That line of reasoning puzzles me, because the anthropic-principle explanation of fine tuning works just fine without MWI: Out of all the conceivable worlds, of course we find ourselves in one that is habitable.
This only works if all worlds that follow the same fundamental theory exist in the same way our local neighborhood exists. If all of space has just one set of constants even though other values would fit the same theory of everything equally well, the anthropic principle does not apply, and so the fact that the universe is habitable is ordinary Bayesian evidence for something unknown going on.
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