In the standard formulation of GR, this is not possible. This is because the metric tensor field is assumed to be everywhere non-degenerate and continuous, and in order to pass from a region of signature (++++) to a region of signature (+---), you need to pass through some regions at which the metric is either degenerate or discontinuous.
As a toy model, one could imagine a metric tensor field of the form
$$ds^2 = dt^2+x (dx^2+dy^2+dz^2)$$
which would have signature (++++) for $x\gt 0$ and $(+---)$ for $x\lt 0$. However, on the hyperplane $x=0$, the metric would have signature $(+000)$ and would therefore be degenerate.
These assumptions are baked into Einstein's equations. Recall that
$$\Gamma^i_{\ \ j k}= \frac{1}{2} g^{i\alpha}(\partial_j g_{\alpha k} + \partial _k g_{\alpha j} - \partial_\alpha g_{jk})$$
and
$$R = g^{\alpha \beta}R_{\alpha \beta}$$
If the metric is not at least continuous, then the $\Gamma$'s are not well-defined (or involve things like delta functions) at the points of metric discontinuity. Furthermore, the components of the inverse metric are the matrix inverse of the components of the metric; if the metric is degenerate, then it is non-invertible, so the "inverse metric" doesn't exist. The entire structure of raising and lowering indices (or at least raising them) depends on the non-degeneracy of the metric, so we must abandon it if we wish to allow metrics with spatially varying signatures.
That being said, apparently the idea of spacetimes which contain multiple regions of different metric signature has been kicked around for a while. A quick search of the arXiv yields this paper, and the subsequent 23 papers which have cited it which you might find interesting. I am nowhere close to an expert on even standard GR, much less fairly dramatic extensions of it, so this is where my ability to speak coherently on the topic ends.
