Rechargeable Zn-air batteries promise safe energy storage. However, they are limited by the redox potential of O₂/O^2- chemistry in an alkaline electrolyte, resulting in low operating voltages and therefore insufficient energy density to compete with lithium-ion batteries. The O₂/O^2- redox potential increases by 0.8 V in an acidic medium, hinting at a way to boost the voltage: an asymmetric electrolyte configuration decoupling acidic and alkaline electrolytes for the air cathode and zinc anode. Such configuration requires a thin and ionically conductive membrane to separate two mutually incompatible electrolytes. Here, we report a Zn ion-exchange membrane with high ionic conductivity of 1.1 mS cm^-1, which prevents acid-base neutralization. The highly reversible O₂/O^2- reaction in the acid is made possible by compositing a cobalt-coordinated porphyrin-based polymeric framework with MXene as a bifunctional electrocatalyst. The asymmetric Zn-air battery operates at voltages up to 1.85 V and cycles for more than 200 h with a material-level energy density of 1350 Wh kg^-1, projected to a high device-level energy density of 50 Wh kg^-1 (coin cell diameter: 20 mm). The asymmetric configuration withstands pressure up to 4 MPa (∼1200 N), demonstrating excellent structural stability for production and practical applications.