We report on the discovery of enormous negative viscosity in a nematic liquid crystal in the presence of turbulence induced by electric fields. As the negative viscosity in this system is so large, we are able to observe several phenomena originating from it. For example, we observe a spontaneous shear flow that rotates the upper disk of a rheometer, as well as the reversal of the rotational direction upon applying an external torque in the opposite direction. Hysteresis loops are also observed in the shear-stress-shear-rate curves, which is reminiscent of those seen for ferromagnetic and ferroelectric materials. The similarities between the phenomena observed for our system and ferroic materials are comprehensively demonstrated, although the two systems are fundamentally different in that the former is out of equilibrium. We elucidate the origin of the negative viscosity and propose a simple model that reproduces the phenomena observed in this active fluid.