  We develop a relativistic (quasi-)hydrodynamic framework, dubbed the
gyrohydrodynamics, to describe fluid dynamics of many-body systems with spin
under strong vorticity based on entropy-current analysis. This framework
generalizes the recently-developed spin hydrodynamics to the regime where the
spin density is at the leading order in derivatives but suppressed by another
small parameter, the Planck constant $\hbar$, due to its quantum nature. Our
analysis shows that the complete first-order constitutive relations of
gyrohydrodynamics involve seventeen transport coefficients and are highly
anisotropic.
