We report on a numerical study of the real-time dynamics of chirally imbalanced lattice Dirac fermions coupled to dynamical electromagnetic field. To this end we use the classical statistical field theory approach, in which the quantum evolution of fermions is simulated exactly, and electromagnetic fields are treated as classical. Motivated by recent experiments on chirally imbalanced Dirac semimetals, we use the Wilson-Dirac lattice Hamiltonian for fermions in order to model the emergent nature of chiral symmetry at low energies. In general, we observe that the backreaction of fermions on the electromagnetic field prevents the system from acquiring large chirality imbalance. In the case of chirality pumping in parallel electric and magnetic fields, electric field is screened by the produced on-shell fermions and the accumulation of chirality is hence stopped. In the case of evolution with initially present chirality imbalance, axial charge tends to decay at the expense of nonzero helicity of electromagnetic field. This decay process, however, shows many unexpected features. In particular, nonzero magnetic helicity is generated due to the suppression, rather than enhancement, of the modes of electromagnetic field with suitable circular polarization. As a result, the energy is transferred from electromagnetic field to fermionic degrees of freedom and not vice versa. We also observe only a rather weak transfer of energy to short-wavelength modes with zero helicity and an even weaker transfer to long-wavelength modes. No signatures of inverse cascade or a turbulent behavior are found. Furthermore, we find that the decay process becomes significantly slower upon moderate decrease of the Fermi velocity from unity, which suggests that the chiral plasma instability might be irrelevant for chirally imbalanced Dirac and Weyl semimetals.