We know that monochromatic lasers produce monochromatic light, i.e., all photons have the same wavelength $\lambda$ (ideally). Coherence, on the other hand, states that the phases of photons are in sync w.r.t. each other. Of course, if the photons had different wavelengths, their phases could not match up due to different wavelength or "repetition cycles". But, if I had a "double wavelength laser" that produced two wavelengths $\lambda_1$ and $\lambda_2=2\cdot \lambda_1$, wouldn't the phases still match up, at least from the point of view of $\lambda_2$? Such a laser could be "coherent", but would definitely not be monochromatic.
In other words: are monochromaticity and coherence two distinct qualities of laser light, or more two very similar qualities? I'm thinking of a hypotehtical "laser" that would emit photons of the same wavelength, but with non-matched photon phases. Of course this wouldn't work because of the disrupted stimulated emission process etc., but it's more of a gedankenexperiment.
For context, I am defining the qualities of laser light, and I'm having an argument with my supervisor about this issue.