We consider a two-dimensional lattice gas model with repulsive nearest- and next-nearest-neighbor interactions that evolves in time according to anisotropic Kawasaki dynamics. The hopping of particles along the principal directions is governed by two heat baths at different temperatures [formula presented] and [formula presented]. The stationary states of this nonequilibrium model are studied using a simple mean-field theory and linear stability analysis. The results are improved by a generalized dynamical mean-field approximation. In the stable ordered state the particles form parallel chains which are oriented along the direction of the higher temperature. In the resulting phase diagram in the [formula presented]-[formula presented] plane the critical temperature curve shows a weak maximum as a function of the parallel temperature which is confirmed by Monte Carlo simulations. Finite-size scaling analysis suggests that the model leaves the equilibrium universality class of the x-y model with cubic anisotropy and is described by the Ising exponents.