We analyze the effect of a small inclination on the well-studied problem of two-dimensional binary fluid convection in a horizontally extended closed rectangular box with a negative separation ratio, heated from below. The horizontal component of gravity generates a shear flow that replaces the motionless conduction state when inclination is not present. This large-scale flow interacts with the convective currents resulting from the vertical component of gravity. For very small inclinations the primary bifurcation of this flow is a Hopf bifurcation that gives rise to chevrons and blinking states similar to those obtained with no inclination. For larger but still small inclinations this bifurcation disappears and is superseded by a fold bifurcation of the large-scale flow. The convecton branches, i.e., branches of spatially localized states consisting of counterrotating rolls, are strongly affected, with the snaking bifurcation diagram present in the noninclined system destroyed already at small inclinations. For slightly larger but still small inclinations we obtain small-amplitude localized states consisting of corotating rolls that evolve continuously when the primary large-scale flow is continued in the Rayleigh number. These localized states lie on a solution branch with very complex behavior strongly dependent on the values of the system parameters. In addition, several disconnected branches connecting solutions in the form of corotating rolls, counterrotating rolls, and mixed corotating and counterrotating states are also obtained.