If you take the Bell test experiment where Alice and Bob perform their measurements at approximately the same time but very far apart, I think you and I both agree that when Alice does a measurement and observes an outcome, she will have locally decohered from the world where she observes the other outcome.
But I don't see why the decoherence necessarily has to happen faster than the speed of the light.
It makes sense that even if Alice decoheres from the world where she observes the other outcome, the outcomes of Bob's measurement are still in a superposition with respect to each Alice (and vice-versa).
And that only when Alices' and Bobs' light cones intersect each other will the Alices decohere from the Bobs in such a way that the resulting worlds will observe the expected correlations (due to how they were entangled or maybe even due to the worlds interfering with each other when their light cones intersect, like what happens in general with the wave function).
I admit I'm not an expert in this area, but is this not possible?