  Flag matroids are combinatorial abstractions of flags of linear subspaces,
just as matroids are of linear subspaces. We introduce the flag Dressian as a
tropical analogue of the partial flag variety, and prove a correspondence
between: (a) points on the flag Dressian, (b) valuated flag matroids, (c) flags
of projective tropical linear spaces, and (d) coherent flag matroidal
subdivisions. We introduce and characterize projective tropical linear spaces,
which serve as a fundamental tool in our proof. We apply the correspondence to
prove that all valuated flag matroids on ground set up to size 5 are
realizable, and give an example where this fails for a flag matroid on 6
elements.
