The following is speculative and without proper foundation. I overwork my geometric intuition here. Perhaps some of the issues can be settled by computation, perhaps others will require better theoretical work.

I suspect that it is necessary for a simplex mesh to include both time and space like edges. The squared edge lengths will thus have both signs. This is not a topological requirement, but a requirement for correct physics. Indeed if there were no space like edges there would be no way for effects to travel travel thru the mesh as fast as they do thru real space. This is the analog to the limits on dt in normal difference equations.

One scheme that attracts me is inspired by a note in Wald’s book concerning sufficient initial value information for propagating forward in time, given empty space.
The following is sufficient to compute forwards or backwards in time from a space like 3D section of the space-time manifold:

Thus two symmetric tensor fields on a 3D manifold suit to start the initial value problem. That the extrinsic curvature is a symmetric (non-positive definite) tensor can be seen as follows. Consider a tangent plane to the 3D section at a point on the section. The departure from that plane of the manifold, in the vicinity of the point is a quadratic function in the coordinates of that plane—a symmetric rank two tensor.

It is clear how to compute the extrinsic curvature when you have nominated a space slice which is a set of tetrahedra all of whose edges are space like. You take one of the triangular faces of one of these tetrahedra and compute “half way” around that triangular bone until you come to the tetrahedron in the slice on the other side of the bone. The cumulative coordinate transformations passing thru the past tense zones, compared with the ???