We investigate the dynamical properties of two mutually delay-coupled semiconductor lasers that are coupled via their optical fields. Because a semiconductor laser is an oscillator that features strong coupling between its amplitude and phase, this system serves as a prototype model of coupled amplitude-phase oscillators. Our main interest here is in the dynamics near and within the locking region where the two lasers emit light of the same frequency. We present experimental observations that give evidence for four qualitatively different dynamical regimes: stable continuous wave emission, oscillations at the laser's characteristic relaxation oscillation frequency, oscillation related to the frequency difference between the two lasers and more complicated dynamics. We characterize and identify these dynamical regimes and analyse them by means of a bifurcation analysis of the corresponding rate equation model with delay. Specifically, we present the underlying bifurcation structure, where the detuning and the pump current are the main bifurcation parameters. The combination of experiment and bifurcation analysis shows how changes in the dynamics arise from the presence of local and global bifurcations near the locking region. © 2009 IOP Publishing Ltd and London Mathematical Society.