I’m not at my normal computer right now, but a few pointers to useful references.
It shows the dual construction quite well.
and my main reference on this was Stefan’s diploma thesis:
Discrete Minimal Surfaces, Koebe Polyhedra, and Alexandrov’s Theorem. Variational Principles, Algorithms, and Implementation. (Diploma Thesis)
Note that this approach works on the principal curvature aligned mesh, then the asymptotic grid would be the diagonals. It looks like what you have is the diagonal grid to start with.
I think I posted on that other thread the definition which takes a starting polyhedron and makes it Koebe, then it has a second stage where you can use the slider to turn one patch of it into the Christoffel dual, which is then discrete minimal.
You can then mirror this patch around to get the rest of the surface by symmetry.
Knowing which polyhedron to use at the start isn’t always immediately obvious though. Those for the standard P G and D surfaces are easy enough to find, but others might take a bit more searching.
If you have a good approximation of the minimal surface to start with, you can maybe run the dual operation the other way (again on a single patch), and see the symmetry from that to figure out what the right polyhedron is.
There’s probably also some easier way I’m missing to find it directly from knowing the symmetry of the minimal surface you’re after.