  Transition metal surfaces catalyse a broad range of thermally-activated
reactions involving carbon-containing-species -- from atomic carbon to small
hydrocarbons or organic molecules, and polymers. These reactions yield
well-separated phases, for instance graphene and the metal surface, or, on the
contrary, alloyed phases, such as metal carbides. Here, we investigate carbon
phases on a rhenium (0001) surface, where the former kind of phase can
transform into the latter. We find that this transformation occurs with
increasing annealing time, which is hence not suitable to increase the quality
of graphene. Our scanning tunneling spectroscopy and reflection high-energy
electron diffraction analysis reveal that repeated short annealing cycles are
best suited to increase the lateral extension of the structurally coherent
graphene domains. Using the same techniques and with the support of density
functional theory calculations, we next unveil, in real space, the symmetry of
the many variants (two six-fold families) of a rhenium surface carbide observed
with diffraction since the 1970s, and finally propose models of the atomic
details. One of these models, which nicely matches the microscopy observations,
consists of parallel rows of eight aligned carbon trimers with a so-called
$(7\times\sqrt{\mathrm{19}})$ unit cell with respect to Re(0001).
