  Swartz proved that any matroid can be realized as the intersection lattice of
an arrangement of codimension one homotopy spheres on a sphere. This was an
unexpected extension from the oriented matroid case, but unfortunately the
construction is not explicit. Anderson later provided an explicit construction,
but had to use cell complexes of high dimensions that are homotopy equivalent
to lower dimensional spheres.
  Using diagrams of spaces we give an explicit construction of arrangements in
the right dimensions. Swartz asked if it is possible to arrange spheres of
codimension two, and we provide a construction for any codimension. We also
show that all matroids, and not only tropical oriented matroids, have a
pseudo-tropical representation.
  We determine the homotopy type of all the constructed arrangements.
