The physical origin of high-velocity cool gas seen in galactic winds remains unknown. Following work by B. Wang, we argue that radiative cooling in initially hot thermally-driven outflows can produce fast neutral atomic and photoionized cool gas. The inevitability of adiabatic cooling from the flow's initial 107–108 K temperature and the shape of the cooling function for T ≲ 107 K imply that outflows with hot gas mass-loss rate relative to star formation rate of $\beta =\dot{M}_{\rm hot}/\dot{M}_\star \gtrsim 0.5$ cool radiatively on scales ranging from the size of the energy injection region to tens of kpc. We highlight the β and star formation rate surface density dependence of the column density, emission measure, radiative efficiency, and velocity. At r cool, the gas produces X-ray and then UV/optical line emission with a total power bounded by ∼10−2 L ⋆ if the flow is powered by steady-state star formation with luminosity L ⋆. The wind is thermally unstable at r cool, potentially leading to a multiphase medium. Cooled winds decelerate significantly in the extended gravitational potential of galaxies. The cool gas precipitated from hot outflows may explain its prevalence in galactic haloes. We forward a picture of winds whereby cool clouds are initially accelerated by the ram pressure of the hot flow, but are rapidly shredded by hydrodynamical instabilities, thereby increasing β, seeding radiative and thermal instability, and cool gas rebirth. If the cooled wind shocks as it sweeps up the circumgalactic medium, its cooling time is short, thus depositing cool gas far out into the halo. Finally, conduction can dominate energy transport in low-β hot winds, leading to flatter temperature profiles than otherwise expected, potentially consistent with X-ray observations of some starbursts.