Broadband atmospheric phase spectra are acquired with a phase-sensitive dual-frequency-comb spectrometer by implementing adaptive compensation for the strong decoherence from atmospheric turbulence. The compensation is possible due to the pistonlike behavior of turbulence across a single spatial-mode path combined with the intrinsic frequency stability and high sampling speed associated with dual-comb spectroscopy. The atmospheric phase spectrum is measured across 2 km of air at each of the 70,000 comb teeth spanning 233 cm(-1) across hundreds of near-infrared rovibrational resonances of CO(2), CH(4), and H(2)O with submilliradian uncertainty, corresponding to a 10(-13) refractive index sensitivity. Trace gas concentrations extracted directly from the phase spectrum reach 0.7 ppm uncertainty, demonstrated here for CO(2). While conventional broadband spectroscopy only measures intensity absorption, this approach enables measurement of the full complex susceptibility even in practical open path sensing.