This is not an answer to your (interesting) question, but instead mentions a possibly related question.

Cardinal, Jean, and Udo Hoffmann. "Recognition and complexity of point visibility graphs." Discrete & Computational Geometry 57.1 (2017): 164-178. (earlier arXiv version.)

Perhaps their ~30 references, or the ~10 subsequent citations, might lead somewhere...?

"We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals."

This is not an answer to your (interesting) question, but instead mentions a possibly related question.

Cardinal, Jean, and Udo Hoffmann. "Recognition and complexity of point visibility graphs." Discrete & Computational Geometry 57.1 (2017): 164-178. (earlier arXiv version.)

Perhaps their ~30 references, or the ~10 subsequent citations, might lead somewhere...?

"We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals."