  Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel
map constructs a version of their row space, whose Pl\"ucker coordinates are
tropical determinants. We explicitly describe the fibers of this map. From the
viewpoint of matroid theory, the tropical Stiefel map defines a generalization
of transversal matroids in the valuated context, and our results are the
valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on
the set of all set families that present a given transversal matroid. We show
that a connected valuated matroid is transversal if and only if all of its
connected initial matroids are. The duals of our results describe complete
stable intersections via valuated strict gammoids.
