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 recentlydeveloped 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 , due to its quantum nature. Our analysis shows that the complete first-order constitutive relations of gyrohydrodynamics involve fourteen transport coefficients and are highly anisotropic.