Ordinary quantum mechanics does not predict that the world splits, no more than does ordinary probability theory.

The zigzag interpretations are entirely relativistic, since the essence of a zigzag interpretation is that you have ordinary space-time with local causality, but you have causal chains that run backwards as well as forwards in time, and a zigzag in time is what gives you unusual spacelike correlation.

A "quantum causal history" (see previous link) is something like a cellular automaton with no fixed grid structure, no universal time, and locally evolving Hilbert spaces which fuse and join.

These three ideas - many worlds, zigzag, QCH - all define research programs rather than completed theories. The latter two are single-world theories and they even approximate locality in spacetime (and not just "in configuration space"). You should think about them some time.