Living nematic is a realization of an active matter combining a nematic liquid crystal with swimming bacteria. The material exhibits a remarkable tendency towards spatio-temporal self-organization manifested in the formation of dynamic textures of self-propelled half-integer topological defects (disclinations or vortices). The well-established and validated model of nematic liquid crystals coupled to the bacterial dynamics is used to describe intricate properties of such a living nematic. The model yielded a testable prediction on the accumulation of bacteria in the cores of 1/2 topological defects and depletion of bacteria in the cores of -1/2 defects. We also studied of such living nematic near normal inclusions, or tactoids, naturally realized in liquid crystals close to the isotropic-nematic (I-N) phase transition. On the basis of the computational analysis, we have established that tactoid's I-N interface spontaneously acquire negative topological charge which is proportional to the tactoid's size and depends on the concentration of bacteria.  The observed negative charging is attributed to the drastic difference in the mobilities of 1/2 and -1/2 topological defects in active systems. The effect is described in the framework of a kinetic theory for point-like weakly-interacting defects with different mobilities.  Our dedicated experiments fully confirmed both theoretical predictions. The results hint into new strategies for control of active matter.