  We numerically study by lattice Boltzmann simulations the rheological
properties of an active emulsion made of a suspension of an active polar gel
embedded in an isotropic passive background. We find that the hexatic
equilibrium configuration of polar droplets is highly sensitive to both active
injection and external forcing and may either lead to asymmetric unidirectional
states which break top-bottom symmetry or symmetric ones. In this latter case,
for large enough activity, the system develops a shear-thickening regime at low
shear rates. Importantly, for larger external forcing a regime with stable
negative effective viscosity is found. Moreover, at intermediate activity a
region of multistability is encountered and we show that a maximum entropy
production principle holds in selecting the most favorable state.
