  We show that the Chow covectors of a linkage matching field define a
bijection between certain degree vectors and lattice points, and we demonstrate
how one can recover the linkage matching field from this bijection. This
resolves two open questions from Sturmfels and Zelevinsky (1993) on linkage
matching fields. For this, we give an explicit construction that associates a
bipartite incidence graph of an ordered partition of a common set to each
lattice point in a dilated simplex. Given a triangulation of a product of two
simplices encoded by a set of spanning trees on a bipartite node set, we
similarly prove that the bijection from left to right degree vectors of the
trees is enough to recover the triangulation. As additional results, we show a
cryptomorphic description of linkage matching fields and characterise the flip
graph of a linkage matching field in terms of its prodsimplicial flag complex.
Finally, we relate our findings to transversal matroids through the tropical
Stiefel map.
