A distributed controller is designed for robust global phase synchronization of a network of uncertain second-order Kuramoto oscillators with a leader system, modeled as a nonlinear autonomous exosystem. The phase angles being elements of the unit circle, we propose an adaptive hybrid strategy based on a hysteresis mechanism to obtain global results despite the well-known topological obstructions. Only an upper bound on the unknown parameters of the oscillators is required to keep the adaptive estimates in a compact set. Since the reference signal is not available to each network node, we design a distributed observer of the leader exosystem. Leveraging the results of hybrid systems theory, including reduction theorems, Lyapunov techniques, and properties of ω-limit sets, we prove robust global asymptotic stability of the closed-loop dynamics, despite the presence of an adaptive control law.