If we put aside the question of dynamics, and just think in terms of a space-time metric, you can certainly have an n-dimensional space in which different regions have different numbers of time dimensions (and in which the light-cone causal structure of ordinary space-time, is replaced by a structure built from multi-time analogues of light cones).
I'm not quite sure what the metric looks like at the boundaries of these regions, but probably it becomes "degenerate" in some way, with some spacelike or timelike distances collapsing to zero.
When it comes to dynamics, it's a lot messier. Even if we don't worry about the coexistence of regions with different numbers of time dimensions, just defining dynamics in a space with multiple time dimensions is problematic. Two of the best-known multi-time frameworks - F-theory and Itzhak Bars's two-time mechanics - actually reduce to single-time physics in practice, as far as I know.
The main problem with multi-time dynamics is simply that dynamics are underdetermined - the initial conditions aren't enough to determine the subsequent evolution. This is not always true, apparently, there are special cases where the technical condition of hyperbolicity is met in such a way that there is a unique multi-time evolution, but I would think that in general, this is not so.
Despite these problems, multi-time physics is probably a valid topic for quantum gravity, because fluctuations of the metric ought to be capable of creating regions with a multi-time metric. Or at least, if your theory of quantum gravity doesn't create such regions, it ought to provide a principled explanation of why it doesn't. Relying on a kind of superselection rule that simply discards multi-time regions because we don't know how to calculate there, seems inadequate to me.
In the case of string theory, along with the F-theory, S-theory approach of adding new time dimensions to M-theory, there is also the application of "timelike dualities" to M-theory which turn some space dimensions into time dimensions. The original big work on this is Chris Hull 1998, and the last major work on the topic that I know about is Dijkgraaf et al 2016.
Quanta Magazine adds that the negative branes of the latter paper may show up as instantons ("instantaneous" transitional objects) in some path integrals. Conceivably this will be string theory's answer to the question of multi-time regions being created by metric fluctuations.
One final comment: you ask about this in the context of eternal inflation - might multi-time regions be created, but then how could they coexist with single-time regions like the one we inhabit? The understanding of eternal inflation in string theory is extremely shaky. Even without the issue of changing time dimensions, the "space" of string vacua is only very partially mapped.
Also, there is a big debate within string theory about how to represent geometries with accelerating expansion - some of the historic examples that were constructed (the "KKLT" model) are now widely believed to be unstable via "brane nucleation". The models of stringy eternal inflation that people have, are incredibly heuristic and only have a minimal connection to actual string theory.
So if I try to sum up (and you should read all of this as "in my opinion"). Multi-time geometries are potentially an issue for any theory of quantum gravity. In string theory, they appear to be related to these negative branes, around which physics is described by one of the timelike duals of M-theory, and which may only have a transient virtual existence as a nonperturbative contribution to a geometric path integral. Or possibly they can have a metastable existence, as a temporary boundary between two space-time regions with different metric signatures (different numbers of time dimensions). Evidently, the right way to think about these things within string theory is not yet figured out; but doing so may be a large part of finally getting string cosmology right.
