  We generalize a recently introduced formulation of relativistic spinful and
vortical fluid to relativistic magnetohydrodynamics (MHD). We refer to it as
the "Spinful-Vortical MHD" (SVMHD). The aim is to scrutinize the interplay
between the vorticity, magnetic field, and spin, which is treated as a quantum
object, in contrast to other formulations of spin hydrodynamics. To this
purpose, we first perform a standard entropy current analysis up to first-order
gradient expansion, $\mathcal{O}\left(\partial\right)$ as well as
$\mathcal{O}\left(\hbar\partial\right)$, where $\hbar$ is the Planck constant.
In contrast to alternative formulations of spin MHD, in the absence of
vorticity, the zeroth-order energy-momentum tensor includes an additional
magneto-vorticity mixed term and reduces, as expected, to the energy-momentum
tensor of MHD. We show that in the first-order of gradient expansion, $36$
dissipative transport coefficients appear. They satisfy certain constraints
that guarantee the positive definiteness of the entropy production rate. We
then modify the formulation of SVMHD by replacing the magnetic part of the
thermal vorticity tensor with its electric part. Carrying out the same analysis
as in the standard formulation, we show that in this case, the first-order
constitutive relations consist of $11$ nondissipative Hall-like coefficients,
apart from $25$ dissipative coefficients. This difference arises from different
behavior of the electric and magnetic part of the thermal vorticity under
time-reversal transformation.
