Light scattering and optical microscopy have been used to measure the morphology as a function of shear rate and composition in polymer blends with viscoelastic asymmetry in the melt components. The blends studied are immiscible mixtures of low-vinyl polybutadiene (PB) and high-vinyl polyisoprene (PI), where the vinyl content strongly influences the rheological properties of the melt. At the temperatures where the optical measurements described here were performed, the PI starts to exhibit an elastic response above a critical shear rate gamma(c), while the PB responds like a viscous fluid up to the highest shear rates of interest. The disparate rheology of the two fluids leads to a rich variety of domain patterns and orientations as the volume fraction of the more elastic component is varied.