  We present the first example of a phase transition in a nonequilibrium
steady-state that can be argued analytically to be first order. The system of
interest is a two-species reaction-diffusion problem whose control parameter is
the total density \rho. Mean-field theory predicts a second-order transition
between two stationary states at a critical density \rho=\rho_c. We develop a
phenomenological picture that, instead, below the upper critical dimension
d_c=4, predicts a first-order transition. This picture is confirmed by
hysteresis found in numerical simulations, and by the study of a
renormalization-group improved equation of state. The latter approach is
inspired by the Weinberg-Coleman mechanism in QED.
