  We introduce a construction of oriented matroids from a triangulation of a
product of two simplices. For this, we use the structure of such a
triangulation in terms of polyhedral matching fields. The oriented matroid is
composed of compatible chirotopes on the cells in a matroid subdivision of the
hypersimplex, which might be of independent interest. In particular, we
generalize this using the language of matroids over hyperfields, which gives a
new approach to construct matroids over hyperfields. A recurring theme in our
work is that various tropical constructions can be extended beyond
tropicalization with new formulations and proof methods.
