  The theory of cold, relativistic, magnetohydrodynamic outflows is generalized
by the inclusion of an intense radiation source. In some contexts, such the
breakout of a gamma-ray burst jet from a star, the outflow is heated to a high
temperature at a large optical depth. Eventually it becomes transparent and is
pushed to a higher Lorentz factor by a combination of the Lorentz force and
radiation pressure. We obtain its profile, both inside and outside the fast
magnetosonic critical point, when the poloidal magnetic field is radial and
monopolar. Most of the energy flux is carried by the radiation field and the
toroidal magnetic field that is wound up close to the rapidly rotating engine.
Although the entrained matter carries little energy, it couples the radiation
field to the magnetic field. Then the fast critical point is pushed inward from
infinity and, above a critical radiation intensity, the outflow is accelerated
mainly by radiation pressure. We identify a distinct observational signature of
this hybrid outflow: a hardening of the radiation spectrum above the peak of
the seed photon distribution, driven by bulk Compton scattering. The
non-thermal spectrum -- obtained by a Monte Carlo method -- is most extended
when the Lorentz force dominates the acceleration, and the seed photon beam is
wider than the Lorentz cone of the MHD fluid. This effect is a generic feature
of hot, magnetized outflows interacting with slower relativistic material. It
may explain why some GRB spectra appear to peak at photon energies above the
original Amati et al. scaling. A companion paper addresses the case of jet
breakout, where diverging magnetic flux surfaces yield strong MHD acceleration
over a wider range of Lorentz factor.
