Sharp bisimulation is a refinement of branching bisimulation, parameterized by a subset of the system's actions, called strong actions. This parameterization allows the sharp bisimulation to be tailored by the property under verification, whichever property of the modal µ-calculus is considered, while potentially reducing more than strong bisimulation. Sharp bisimulation equivalence is a congruence for process algebraic operators such as parallel composition, hide, cut, and rename, and hence can be used in a compositional verification setting. In this paper, we prove that sharp bisimulation equivalence is also a congruence for action priority operators under some conditions on strong actions. We compare sharp bisimulation with orthogonal bisimulation, whose equivalence is also a congruence for action priority. We show that, if the internal action τ neither gives priority to nor takes priority over other actions, then the quotient of a system with respect to sharp bisimulation equivalence (called sharp minimization) cannot be larger than the quotient of the same system with respect to orthogonal bisimulation equivalence. We then describe a signature-based partition refinement algorithm for sharp minimization, implemented in the BCG_MIN and BCG_CMP tools of the CADP software toolbox. This algorithm can be adapted to implement orthogonal minimization. We show on a crafted example that using compositional sharp minimization may yield state space reductions that outperform compositional orthogonal minimization by several orders of magnitude. Finally, we illustrate the use of sharp minimization and priority to verify a bully leader election algorithm.