Suppose we perform an experiment where, based on the measured spin value, we move some macroscopic object with detectable gravity in opposite directions. In the Newtonian background spacetime approach there is no issue with MWI, as both branches live on the same spacetime. In a full GR case, however, the spacetime itself must decohere into different branches, or else we could detect the interaction between different branches gravitationally (I don't know if this has been tested, but it would be extremely surprising if detected).

I can't get past the paywall, but I think this is what Page and Geilker do in "Indirect Evidence for Quantum Gravity".