Copolymers and stirring are commonly used to produce fine dispersions of immiscible homopolymers. Recent experiments call into question the classical view that copolymers promote the mixing by reducing the interfacial tension, suggesting rather that copolymers induce repulsive interactions between droplets and thus inhibit collision-induced coalescence events. We present a dynamical theory of the breakup and coalescence of polymer droplets in a mixing shear how including hydrodynamic and repulsive interactions between droplets. We find that a low surface coverage of copolymers, of the order of one chain per square radius of gyration, is sufficient to inhibit collisions between submicron-sized droplets, while giving a negligible reduction in interfacial tension.