  Two formulations of relativistic hydrodynamics of particles with spin 1/2 are
compared. The first approach, dubbed the canonical one, uses expressions for
the energy-momentum and spin tensors that have properties that follow a direct
application of Noether's theorem, which yields a totally antisymmetric spin
tensor. The other one is based on a simplified form of the spin tensor and is
commonly used in the current literature under the name of a phenomenological
approach. We show that these two frameworks are equivalent, i.e., they can be
directly connected by a suitably defined pseudogauge transformation, only if
the first framework is initially improved by a suitable modification of the
energy-momentum tensor (addition of a divergence-free term that cannot be
interpreted as a pseudogauge). Our analysis uses arguments related to the
positivity of entropy production. The latter turns out to be equivalent for the
improved canonical and phenomenological frameworks.
