  We consider a finite one-dimensional totally asymmetric simple exclusion
process (TASEP) with four types of particles, $\{1,0,\bar{1},*\}$, in contact
with reservoirs. Particles of species $0$ can neither enter nor exit the
lattice, and those of species $*$ are constrained to lie at the first and last
site. Particles of species $1$ enter from the left reservoir into either the
first or second site, move rightwards, and leave from either the last or
penultimate site. Conversely, particles of species $\bar{1}$ enter from the
right reservoir into either the last or penultimate site, move leftwards, and
leave from either the first or last site. This dynamics is motivated by a
natural random walk on the Weyl group of type D. We compute the exact
nonequilibrium steady state distribution using a matrix ansatz building on
earlier work of Arita. We then give explicit formulas for the nonequilibrium
partition function as well as densities and currents of all species in the
steady state, and derive the phase diagram.
