Atmospheric water harvesting is urgently needed given increasing global water scarcity. Current sorbent-based devices that cycle between water capture and release have low harvesting rates. We envision a radically different multi-material architecture with segregated and simultaneous capture and release. This way, proven fast-release mechanisms that approach theoretical limits can be incorporated; however, no capture mechanism exists to supply liquid adequately for release. Inspired by tree frogs and airplants, our capture approach transports water through a hydrogel membrane ``skin'' into a liquid desiccant. We report an extraordinarily high capture rate of 5.50 $\text{kg}\,\text{m}^{-2}\,\text{d}^{-1}$ at a low humidity of 35%, limited by the convection of air to the device. At higher humidities, we demonstrate up to 16.9 $\text{kg}\,\text{m}^{-2}\,\text{d}^{-1}$, exceeding theoretical limits for release. Simulated performance of a hypothetical one-square-meter device shows that water could be supplied to two to three people in dry environments. This work is a significant step toward providing new resources to water-scarce regions.