Phospholipid membranes surround the cell and its internal organelles, and their multicomponent nature allows the formation of domains that are important in cellular signalling, the immune system, and bacterial infection. Cytoplasmic compartments are also created by the phase separation of intrinsically disordered proteins into biomolecular condensates. The ubiquity of lipid membranes and protein condensates raises the question of how three-dimensional droplets might interact with two-dimensional domains, and whether this coupling has physiological or pathological importance. Here, we explore the equilibrium morphologies of a dilute phase of a model disordered protein interacting with an ideal-mixing, two-component lipid membrane using coarse-grained molecular simulations. We find that the proteins can wet the membrane with and without domain formation, and form phase separated droplets bound to membrane domains. Results from much larger simulations performed on a novel non-von-Neumann compute architecture called POETS, which greatly accelerates their execution compared to conventional hardware, confirm the observations. Reducing the wall clock time for such simulations requires new architectures and computational techniques. We demonstrate here an inter-disciplinary approach that uses real-world biophysical questions to drive the development of new computing hardware and simulation algorithms.