Dual-comb spectroscopy allows for the rapid, multiplexed acquisition of high-resolution spectra without the need for moving parts or low-resolution dispersive optics. This method of broadband spectroscopy is most often accomplished via tight phase locking of two mode-locked lasers or via sophisticated signal processing algorithms, and therefore, long integration times of phase coherent signals are difficult to achieve. Here we demonstrate an alternative approach to dual-comb spectroscopy using two phase modulator combs originating from a single continuous-wave laser capable of > 2 hours of coherent real-time averaging. The dual combs were generated by driving the phase modulators with step-recovery diodes where each comb consisted of > 250 teeth with 203 MHz spacing and spanned > 50 GHz region in the near-infrared. The step-recovery diodes are passive devices that provide low-phase-noise harmonics for efficient coupling into an enhancement cavity at picowatt optical powers. With this approach, we demonstrate the sensitivity to simultaneously monitor ambient levels of CO2, CO, HDO, and H2O in a single spectral region at a maximum acquisition rate of 150 kHz. Robust, compact, low-cost and widely tunable dual-comb systems could enable a network of distributed multiplexed optical sensors.