  We study the non-equilibrium steady states in a totally-asymmetric
simple-exclusion process with periodic boundary conditions, also incorporating
(i) an extra (nearest-neighbour) repulsive interaction and (ii) hopping rates
characterized by a smooth spatial inhomogeneity. We make use of a generalized
mean-field approach (at the level of nearest-neighbour pair clusters), in
combination with kinetic Monte Carlo simulations. It turns out that the
so-called shock phase can exhibit a lot of qualitatively different subphases,
including multiple-shock phases, and a minimal-current shock phase. We argue
that the resulting, considerably rich phase diagram should be relatively
insensitive to minor details of either interaction or spatial inhomogeneity. As
a consequence, we also expect that our results help elucidate the nature of
shock subphases detected in previous studies.
