We consider a system of two Gross–Pitaevskii (GP) equations, in the presence of an optical-lattice (OL) potential, coupled by both nonlinear and linear terms. This system describes a Bose–Einstein condensate (BEC) composed of two different spin states of the same atomic species, which interact linearly through a resonant electromagnetic field. In the absence of the OL, we find plane-wave solutions and examine their stability. In the presence of the OL, we derive a system of amplitude equations for spatially modulated states, which are coupled to the periodic potential through the lowest order subharmonic resonance. We determine this averaged system’s equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions with direct simulations of the coupled GP equations. We find that symmetric (equal-amplitude) and asymmetric (unequal-amplitude) dual-mode resonant states are, respectively, stable and unstable. The unstable states generate periodic oscillations between the two condensate components, which are possible only because of the linear coupling between them. We also find four-mode states, but they are always unstable. Finally, we briefly consider ternary (three-component) condensates.