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Lipid rafts are nanoscopic assemblies of sphingolipids, cholesterol, and specific membrane proteins. They are believed to underlie the experimentally observed lateral heterogeneity of eukaryotic plasma membranes and implicated in many cellular processes, such as signaling and trafficking. Ternary model membranes consisting of saturated lipids, unsaturated lipids, and cholesterol are common proxies because they exhibit phase coexistence between a liquid-ordered (lo) and liquid-disordered (ld) phase and an associated critical point. However, plasma membranes are also asymmetric in terms of lipid type, lipid abundance, leaflet tension, and corresponding cholesterol distribution, suggesting that rafts cannot be examined separately from questions about elasticity, curvature torques, and internal mechanical stresses. Unfortunately, it is challenging to capture this wide range of physical phenomenology in a single model that can access sufficiently long length- and time scales. Here we extend the highly coarse-grained Cooke model for lipids, which has been extensively characterized on the curvature-elastic front, to also represent raft-like lo/ld mixing thermodynamics. In particular, we capture the shape and tie lines of a coexistence region that narrows upon cholesterol addition, terminates at a critical point, and has coexisting phases that reflect key differences in membrane order and lipid packing. We furthermore examine elasticity and lipid diffusion for both phase separated and pure systems and how they change upon the addition of cholesterol. We anticipate that this model will enable significant insight into lo/ld phase separation and the associated question of lipid rafts for membranes that have compositionally distinct leaflets that are likely under differential stress—like the plasma membrane.