First, I give an absurd example.
A conscious observer lives on Earth at time t. A light-year away, at a space-like separation, a nuclear bomb chain reaction goes off. A nuclear bomb chain reaction is quantum all the way and leads to a splitting into expo-exponentially many parallel worlds. In one reference frame, the explosion happens months after the observer at time t. Let's say at time t, we have N copies of the observer out there distributed across N parallel worlds. In another boosted reference frame, the explosion happens before the observer at time t. What this means is, at time t, the observer is split into N times expo-exponentially many copies, and for each version of the observer, there are expo-exponentially many exactly identical copies lying about. By anthropic reasoning, and anthropic reasoning is necessary in the many worlds interpretation, the latter reference frame would be preferred by an expo-exponential factor in complete violation of Lorentz invariance.
The only way to save the many worlds interpretation from this absurdity is to find a manifestly Lorentz invariant version of splitting. How can this be done?