We propose an experimental and theoretical study of the dynamical regimes observed when the two modes of a two-frequency solid-state laser are coupled by frequency-shifted optical feedback. The detuning associated to the feedback is close to the frequency of the laser relaxation oscillations. Special attention is devoted to the dynamics of the phase of the beat note between the modes relative to the phase of an external radio-frequency reference. In particular, we analyze in detail the transition from phase locking to the external reference to phase drift. This transition occurs through a window of frequency locking without phase locking [Thévenin, Romanelli, Vallet, Brunel, and Erneux, Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.107.104101 107, 104101 (2011)]. Furthermore, a large variety of dynamical behaviors, such as weak intensity modulation, self-pulsing, and chaos, are experimentally found. These results are successfully reproduced by a theoretical model based on coupled rate equations. The extension of the model to the description of other active media such as semiconductor lasers is briefly outlined.