We present an experimental and theoretical study of multistability of a single-mode laser subject to feedback through phase tuning and amplifier sections integrated on the same chip. Closely above threshold, a regime of tristability of continuous-wave (CW) states is found for multiple ranges of amplifier and phase currents. The separation between the tristable wavelengths agrees with the channel spacing of dense wavelength multiplexing in the C band of optical communication making the device interesting for ternary logic applications. Complementary theoretical investigations in the framework of the paradigmatic Lang-Kobayashi model provide a consistent understanding of the experimental findings and additionally yield an analytic formula expressing the maximum number of coexisting stable CW states by the linewidth-enhancement factor a.  Tristability belongs to the a range from 5 to 8 in good agreement with experiment.