Since it was first discussed by Baxter in 1970, the three coloring model has been studied in several contexts, from frustrated magnetism to superconducting devices and glassiness. In presence of interactions, when the model is no longer exactly soluble, it was already observed that the phase diagram is highly nontrivial. Here we discuss the generic case of "color-dependent" nearest-neighbor interactions between the vertex chiralities. We uncover different critical regimes merging into one another: c=1/2 free fermions combining into c=1 free bosons; c=1 free bosons combining into c=2 critical loop models; as well as three separate c=1/2 critical lines merging at a supersymmetric c=3/2 critical point. When the three coupling constants are tuned to equal one another, transfer-matrix calculations highlight a puzzling regime where the central charge appears to vary continuously from 3/2 to 2.