Soliton microcombs provide a versatile platform for realizing fundamental studies and technological applications. To be utilized as frequency rulers for precision metrology, soliton microcombs must display broadband phase coherence, a parameter characterized by the optical phase or frequency noise of the comb lines and their corresponding optical linewidths. Here, we analyze the optical phase-noise dynamics in soliton microcombs and show that, because of the Raman self-frequency shift, the fundamental linewidth of some of the comb lines can, surprisingly, be narrower than the linewidth of the pump laser. This work elucidates information about the ultimate limits in phase coherence of soliton microcombs and illustrates a new strategy for the generation of spectrally coherent light on chip.