  We study a one-dimensional totally asymmetric exclusion process with random
particle attachments and detachments in the bulk. The resulting dynamics leads
to unexpected stationary regimes for large but finite systems. Such regimes are
characterized by a phase coexistence of low and high density regions separated
by domain walls. We use a mean-field approach to interpret the numerical
results obtained by Monte-Carlo simulations and we predict the phase diagram of
this non-conserved dynamics in the thermodynamic limit.
