We study the evolution of binary mixtures far from equilibrium, and show that the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures. We describe a model binary system in a Hele-Shaw cell using a phase-field approach with explicit dependence of both phase fraction and mass concentration. When the viscosity contrast between phases is large (as is the case for gas and liquid phases), an imposed background flow leads to viscous fingering, phase branching, and pinch off. This dynamic flow disorder limits phase growth and arrests thermodynamic coarsening. As a result, the system reaches a regime of statistical steady state in which the binary mixture is permanently driven away from equilibrium. ; United States. Dept. of Energy (CAREER Award DE-SC0003907) ; United States. Dept. of Energy (Grants DESC0009286 and DE-FE0013999) ; Spain. Ministerio de Economia y Competividad (Ramon y Cajal Fellowship)