We contrast the evolution of gravitational binary mergers for the two cases of a black hole and a normal neutron star, and a black hole and a self-bound strange quark matter star. In both cases inspiral continues until the Roche limit is reached, at which point it is expected that stable mass transfer to the balck hole ensues. Whereas a neutron star would then outspiral, the strange quark matter star barely moves out at all. Eventually, the strange quark star loses all its mass to the black hole, as compared to the neutron star whose outspiral continues until stable mass transfer terminates and inspiral resumes. These scenarios result in distinctly different gravitational wave signatures.