We consider two-phase fluid deformable surface models with a constant enclosed volume. Such models consist of incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces and a constraint on the enclosed volume. The highly nonlinear model accounts for the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics and allows to model biomembranes, or at least model systems of biomembranes, such as vesicles. We numerically explore the effect of hydrodynamics on bulging and furrow formation and study the effect of a phase-dependent bending and Gaussian bending rigidity on the dynamics and the equilibrium configuration. The numerical approach builds on a Taylor-Hood element for the surface Navier–Stokes part, a semi-implicit approach for the surface Cahn-Hilliard part, higher order surface parametrizations, appropriate approximations of the geometric quantities, mesh redistribution and an iterative approach to deal with the non-local constraint on the enclosed volume. We demonstrate convergence properties.