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Nice! Thank you. It looks like just blow-ups at smooth centers are enough, which is surprising to me. Do you know if there is a version of this without assuming projectiveness?
@JasonStarr I see what you're trying to say and I agree that this point of view should lead to counter-examples. Still, in this specific example $X'$ is still "toric equivalent" to X, by the sequence of maps $\mu, \nu$. Your argument proves is that $X'$ and $X$ don't have a common locally toric refinement.