  We construct a one-dimensional totally asymmetric simple exclusion process
(TASEP) on a ring with two segments having unequal hopping rates, coupled to
particle non-conserving Langmuir kinetics (LK) characterized by equal
attachment and detachment rates. In the steady state, in the limit of competing
LK and TASEP, the model is always found in states of phase coexistence. We
uncover a nonequilibrium phase transition between a three-phase and a two-phase
coexistence in the faster segment, controlled by the underlying inhomogeneity
configurations and LK. The model is always found to be half-filled on average
in the steady state, regardless of the hopping rates and the
attachment/detachment rate.
