Do you see how a nondeterministic Turing machine could model that?

If so ... ... ...

Suppose in QM you have a wavefunction which recognizably evolves into a superposition of wavefunctions. I'll write that psi0, the initial wavefunction, becomes m.psi' + n.psi'', where m and n are coefficients, and psi' and psi'' are basis wavefunctions.

Something slightly analogous to the MWI interpretation of this, could be seen in a Turing machine which started with one copy of a bitstring, PSI0, and which replaced it with M copies of the bitstring PSI' and N copies of the bitstring PSI''. That would be a deterministic computation which replaces one world, the single copy of PSI0, with many worlds, the multiple copies of PSI' and PSI''.

So it's straightforward enough for a deterministic state machine to invent rules corresponding to a proliferation of worlds. In fact, in the abstract theory of computation, this is one of the standard ways to model nondeterministic computation - have a deterministic computation which deterministically produces all the possible paths that could be produced by the nondeterministic process.

However, the way that QM works, and thus the way that a MWI theory would have to work, is rather more complicated, because the coefficients are complex numbers, the probabilities (which one might suppose correspond to the number of copies of each world) are squares of the absolute values of those complex numbers, and probability waves can recombine and destructively interfere, so you would need worlds / bitstrings to be destroyed as well as created.

In particular, it seems that you couldn't reproduce QM with a setup in which the only fact about each world / bitstring was the number of current copies - you need the "phase information" (angle in the complex plane) of the complex numbers, in order to know what the interference effects are. So your Turing machine's representation of the state of the multiverse would be something like:

(complex coefficient associated with the PSI' worlds) (list of M copies of the PSI' bitstring); (complex coefficient associated with the PSI'' worlds) (list of N copies of the PSI'' bitstring) ; ...

and the "lists of copies of worlds" would all be dynamically irrelevant, since the dynamics comes solely from recombining the complex numbers at the head of each list of copies. At each timestep, the complex numbers would be recomputed, and then the appropriate number of world-copies would be entered into each list.

But although it's dynamically irrelevant, that list of copies of identical worlds is still performing a function, namely, it's there to ensure that there actually are M out of every (M+N) observers experiencing worlds of type PSI', and N out of every (M+N) observers experiencing worlds of type PSI''. If the multiverse representation was just

(complex coefficient of PSI' world) (one copy of PSI' world) ; (complex coefficient of PSI'' world) (one copy of PSI'' world) ; ...

then all those complex numbers could still evolve according to the Schrodinger equation, but you would only have one observer seeing a PSI' world, and one observer seeing a PSI'' world, and this is inconsistent with observation, where we see that some quantum events are more probable than others.

This is the well-known problem of recovering the Born probabilities, or justifying the Born probability rule - mentioned in several places in the QM Sequence - but expressed in the unusual context of bit-strings on a Turing tape.

(Incidentally, I have skipped over the further problem that QM uses continuous rather than discrete quantities, because that's not a problem of principle - you can just represent the complex numbers the way we do on real computers, to some finite degree of binary precision.)