This work proposes a distributed control strategy for the robust global leader-follower phase synchronization of Kuramoto oscillators with inertia. For a convenient design, the phase angles are represented as elements of the unit circle. In particular, we exploit a "half-angle" representation inspired by unit quaternions. The ensuing non-Euclidean state space poses some challenges for robust global stabilization, which can be conveniently overcome with dynamic hybrid feedback. For this reason, we propose a hybrid solution obtained by combining a distributed observer with local hysteresis-based tracking controllers. The overall closed-loop system is analyzed through reduction theorems and Lyapunov-based arguments.