  We show that uniform lattices of isometries of products of real hyperbolic
spaces act properly discontinuously and cocompactly on a median space. For
lattices in products of at least two factors, this is the strongest degree of
compatibility possible with the median geometry. Our theorem is also relevant
for potential Rips-type theorems for median spaces. The result follows from an
analysis of a quasification of median geometry that provides a geometric
characterization of spaces at finite Hausdorff distance from a median space. We
explain how the case of complex hyperbolic metric spaces is different, and that
such spaces cannot be at finite Hausdorff distance from a median space.
