  We discuss a puzzle in relativistic spin hydrodynamics; in the previous
formulation the spin source from the antisymmetric part of the canonical
energy-momentum tensor (EMT) is crucial. The Belinfante improved EMT is
pseudo-gauge transformed from the canonical EMT and is usually a physically
sensible choice especially when gauge fields are coupled as in
magnetohydrodynamics, but the Belinfante EMT has no antisymmetric part. We find
that pseudo-transformed entropy currents are physically inequivalent in
nonequilibrium situations. We also identify a current induced by the spin
vorticity read from the Belinfante symmetric EMT.
