The new nonlinear axionically extended version of the general relativistic magnetohydrodynamics is formulated. The self-consistent formalism of this theory is based on the introduction into the Lagrangian of the new unfied scalar invariant, which is quadratic in the Maxwell tensor, and contains two periodic functions of the pseudoscalar (axion) field. The constructed unified invariant and the elaborated nonlinear theory as a whole, are invariant with respect to two symmetries: first, the discrete symmetry associated with the properties of the axion field; second, the Jackson's SO(2) type symmetry intrinsic for the electromagnetism. The subsystem of the master equations, which describes the velocity four-vector of the hydrodynamic flow, is constructed in the framework of Eckart's theory of viscous heat-conducting fluid. The axionically extended nonlinear Faraday, Gauss and Ampere equations are supplemented by the ansatz about the large electric conductivity of the medium, which is usually associated with vanishing of the electric field. We have suggested two essentially new nonlinear models, in the framework of which the anomalous electric conductivity is being compensated by the appropriate behavior of the finite pseudoscalar (axion) field, providing the electric field in the magnetohydrodynamic flow to be finite (either to be proportional to the magnetic field, or to the angular velocity of the medium rotation).