  We present the general relativistic electrodynamics and magnetohydrodynamics
with a helically coupled scalar field. We consider three component system with
the fluid, scalar field and electromagnetic fields with the helical coupling.
We derive three exact formulations: the covariant formulation, the ADM
formulation, and the fully nonlinear and exact perturbation formulation. We
also derive the weak-gravity limit with fully relativistic fluid and fields.
The latter two formulations are presented in the cosmological context.
