See my response to Oscar. Branches (according to Oxford) are like the blobs in configuration space (or like the ink blots on Emile's illustration!), they're an approximate construct that depends on definition. Which is an absolutely untenable position, if you want to identify the observed portion of reality with a branch, because the observed portion of reality does not owe its existence to our definitions.

If people cared about this issue of vagueness, they could seek objective definitions of world (apart from "a point in configuration space"). For example, they could look at maxima, minima, and other turning points in the wavefunction. The blob could be exactly identified by the maximum that it contains, rather than fuzzily identified by its edges. There might be a dual description of the wavefunction in terms of a topological object in configuration space, rather than a complex-valued function, in which specifying the maxima, minima, etc., carries all the same information. That sort of investigation would be much healthier.