Eukaryotic lipid membranes are both compositionally complex and strongly asymmetric. Preferential lipid interactions enable coexistence between two fluid phases and an associated critical point, while bilayer asymmetry leads to leaflet-specific values for many observables-most saliently composition, but also a difference in leaflet tensions, for which we introduced the term "differential stress." Lipid mixing thermodynamics has been extensively studied, notably in idealized ternary model systems, and interest in asymmetry has grown significantly in the past decade, but their interplay remains poorly understood. Here we propose a conceptual framework for the thermodynamics of asymmetric ternary lipid membranes. Cholesterol emerges as an essential actor playing two different roles: first, it controls lipid mixing
second, it couples the compositional phase points of the two leaflets by achieving chemical equilibrium between them. Since differential stress can squeeze cholesterol from one leaflet into the other, this couples mechanical properties such as lateral stresses and curvature torques directly to mixing thermodynamics. Using coarse-grained simulations, we explore implications for leaflet coexistence, mechanical stability of giant vesicles, and differential stress driven phase segregation in a single leaflet. We hope this framework enables a fresh look at some persistent puzzles in this field, most notably the elusive nature of lipid rafts.