Is there a manifestly Lorentz invariant formulation of the many worlds interpretation? 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?
 A: In the so-called "many worlds interpretation," the "splitting of worlds" is not an actual physical process. In this interpretation, all there is is the wavefunction of the Universe, and all it ever does is evolve smoothly according to the appropriate equation of motion. The whole theory is manifestly Lorentz-invariant from end to end. 
The "standard" (Copenhagen) interpretation is the one that's not manifestly Lorentz-invariant, since a measurement at one place causes "collapse" everywhere all at once.
A: In MWI, "splittings" are local. When the nuclear bomb goes off a light year away, the observer here on Earth does not "split" instantaneously. The explosive remnants of the bomb expands in a bubble from the bomb at a speed not greater than the speed of light, and only when one of the remnants reach the Earth-bound observer does the observer "split". This locality of "splittings" is manifestly Lorentz invariant. Moreover, the observer does not "split" into "doubly exponentially" many copies because only a few remnants of the bomb can ever reach him. He only "splits" a few more times.
