Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Plücker 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.