  We study the covering of the plane by non-overlapping rhombus tiles, a
problem well-studied only in the limiting case of dimer coverings of regular
lattices. We go beyond this limit by allowing tiles to take any position and
orientation on the plane, to be of irregular shape, and to possess different
types of attractive interactions. Using extensive numerical simulations we show
that at large tile densities there is a phase transition from a fluid of
rhombus tiles to a solid packing with broken rotational symmetry. We observe
self-assembly of broken-symmetry phases, even at low densities, in the presence
of attractive tile-tile interactions. Depending on tile shape and interactions
the solid phase can be random, possessing critical orientational fluctuations,
or crystalline. Our results suggest strategies for controlling tiling order in
experiments involving `molecular rhombi'.
