We study, via both experiment and theory, localization of longitudinal-acoustic waves scattered from sites supporting transverse (shear) modes. The experimental system consists of a polymer melt solidifying by the growth of spherical semicrystalline nuclei. We excite this system with acoustic plane waves and measure the transmitted signal. For sufficiently high excitation frequencies we find renormalization of the sound speed and intense absorption peaks over a very narrow range of wave number. These data can be consistently interpreted as signs of localization within the sample. Standard theory, however does not predict localization in this system, since the longitudinal velocity in the scatterers is faster than that in the liquid. However, the solid scatterers support shear modes, which can significantly modify their scattering characteristics. We extend the theory of localization to allow for scatterers supporting shear. This model predicts shear-induced localization in the system we have studied.