We investigate the dynamics of asymmetrically coupled semiconductor lasers on photonic integrated circuits in experiment and theory. The experimental observations are explained using a rate-equation model for coupled lasers incorporating a saturable coupling waveguide. We perform a bifurcation analysis of the coupled laser dynamics, focusing on the effects of the coupling phase and the dynamical difference between passive and saturable coupling waveguides. For a passive waveguide, we find a bifurcation scenario closely resembling the well-known optical injection setup, which is largely insensitive to the coupling phase. When the coupling waveguide is saturable, the dynamics become increasingly complex and unpredictable, with a strong phase-dependence. Our results show the possibility of a simple layout for reproducible laser dynamics on a chip.