Microswimmers in a fluid are an example of an active suspension whereby the system is driven out of equilibrium though the interaction of the microswimmers with their surrounding environment. Here, the authors study the orientational microstructure of active suspensions in a viscoelastic fluid and show how the activity of the microswimmers can alter the bulk properties. Microswimmer suspensions in Newtonian fluids exhibit unusual macroscale properties, such as a superfluidic behavior, which can be harnessed to perform work at microscopic scales. Since most biological fluids are non-Newtonian, here we study the rheology of a microswimmer suspension in a weakly viscoelastic shear flow. At the individual level, we find that the viscoelastic stresses generated by activity substantially modify the Jeffery orbits well-known from Newtonian fluids. The orientational dynamics depends on the swimmer type; especially pushers can resist flow-induced rotation and align at an angle with the flow. To analyze its impact on bulk rheology, we study a dilute microswimmer suspension in the presence of random tumbling and rotational diffusion. Strikingly, swimmer activity and its elastic response in polymeric fluids alter the orientational distribution and substantially amplify the swimmer-induced viscosity. This suggests that pusher suspensions reach the superfluidic regime at lower volume fractions compared to a Newtonian fluid with identical viscosity.