Your basis is just a mathematical tool, not a physical property of the system.
I agree- this means for many worlds to make predictions, we need very much for the number of 'branches' (I shall try and consistently use branches from now on, as opposed to blobs, or worlds,etc),however we define them, to be basis-independent. So I was asking- is it?
I shall answer my own question- if you have large objects that can cause decoherence (i.e. density matrices evolve until they are diagonal) then yes- branches are independent, as I've defined them. It is in this sense that decoherence solves the preferred basis problem.
HOWEVER- you aren't guaranteed to ever form those initial decoherent objects. Barvinsky and Kamenschchik (Physical Review D 52, 743-757, I apologize its behind a paywall) showed something neat- if you don't pick specific initial conditions for your universal wavefunction, then you may never get any 'classical' branches of the wavefunction. For other choices of initial condition, branches can evolve from 'classical' to not, very quickly. The non-classical branches ARE basis dependent, and this is problematic- without a unique decomposition, we can't make consistent predictions!
This is very much related to the initial conditions issue in Bohm's theory- which shouldn't be surprising since both interpretations strive to match the same set of predictions (standard quantum).
The only interpretations that avoid this particularly weird issue are ensemble and collapse interpretations (see GRW or one of the many other objective/stochastic collapse theories). In ensembles, this is because the wavefunction isn't "real" in the same way, and we use it only to predict probability distributions. In collapse, this is because collapse is guaranteed to create a diagonal (one entry) density matrix.
I did try earlier in the thread to move on from the multiplicity of worlds arguments, but then people asked more questions.