Laser interferometers are the core measurement tool in gravitational wave observatories. An important factor that can limit the performance is the relative power instability of the laser, a problem often called relative intensity noise (RIN). But exactly how this influences the interferometer performance is not completely understood. Therefore in this paper we analyze laser RIN coupling into the phase readout in balanced and unbalanced heterodyne interferometers. We describe the coupling theoretically, then simulate and finally measure it. Our results reveal a combination of RIN contributions from the heterodyne frequency and twice the heterodyne frequency in the interferometric phase readout. We also show that when an additional, correlated reference measurement is subtracted the combined coupling factor depends on the differential phase between the two measurements and thus can be minimized. Our results have implications for noise models in future space-based gravitational wave observatories like Laser Interferometer Space Antenna, where RIN-to-phase coupling arises directly and is modulated via spacecraft jitter, testmass position and orientation.