  An investigation of the two-dimensional Widom-Rowlinson lattice gas under an
applied drive uncovered a remarkable non-equilibrium steady state in which
uniform stripes (reminiscent of an equilibrium lamellar phase) form
perpendicular to the drive direction [R. Dickman and R. K. P. Zia, Phys. Rev. E
97, 062126 (2018)]. Here we study this model at low particle densities in two
and three dimensions, where we find a disordered phase with a characteristic
length scale (a "microemulsion") along the drive direction. We develop a
continuum theory of this disordered phase to derive a coarse-grained
field-theoretic action for the non-equilibrium dynamics. The action has the
form of two coupled driven diffusive systems with different characteristic
velocities, generated by an interplay between the particle repulsion and the
drive. We then show how fluctuation corrections in the field theory may
generate the characteristic features of the microemulsion phase, including a
peak in the static structure factor corresponding to the characteristic length
scale. This work lays the foundation for understanding the stripe phenomenon
more generally.
