  The inhomogeneous two-species TASEP on a ring is an exclusion process that
describes particles of different species hopping clockwise on a ring with
parameters giving the hopping rates for different species. We introduce a
combinatorial object that we call \emph{toric rhombic alternative tableaux},
which are certain fillings of tableaux on a triangular lattice tiled with
rhombi, and are in bijection with the well-studied \emph{multiline queues} of
Ferrari and Martin. Using the tableaux, we obtain a formula for the stationary
probabilities of this TASEP, which specializes to results of Ayyer and
Linusson. We obtain, in addition, an explicit determinantal formula for these
probabilities, and define a Markov chain on the tableaux that projects to the
two-species TASEP on a ring.
