Summary form only given. Bounded-phase dynamics, also called phase-trapping or frequency-locking without phase-locking, is a synchronization regime found when a phase-locked state bifurcates via a Hopf bifurcation to self-sustained phase and amplitude oscillations. After the instability, the coupled oscillators are not phase-locked anymore, yet they maintain frequency-locking and synchronization [1]. This phenomenon is rather universal, and has been demonstrated for instance in semiconductor and solid-state lasers, in hydrodynamics and in nanomechanical resonators. Recently, the relevance of these underdamped and self-sustained phase oscillations has also started to be investigated for the synchronization of quantum systems [2].To date, observations of the bounded-phase regime have been limited to smooth, supercritical bifurcations. The aim of this contribution is to present an experiment in which a phase-locked state undergoes a subcritical bifurcation leading directly to a chaotic state. We demonstrate the existence of a regime characterized by chaotic phase and amplitude oscillations, and prove that frequency-locking can be maintained, i.e. that bounded-phase synchronization is possible also in the presence of chaotic dynamics. Furthermore, by measuring the phase noise spectrum of the oscillator, we show that the beat-note signal still keeps a good spectral purity at low frequencies, in spite of the chaotic phase oscillations.