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An astounding variety of cellular contexts converge to the process of liquid-liquid phase separation for the creation of new functional levels of organization. But the kinetic pathways by which intracellular phase separation proceeds – typically in physically confined and macromolecularly crowded volumes of topologically closed cellular and intracellular compartments –remain incompletely understood. Here, we monitor the dynamics of liquid-liquid phase separation of mixtures of phase-separating polymers (i.e., polyethyleneglycol and dextran) inside all-synthetic, cell-sized giant unilamellar vesicles in real-time. We dynamically trigger phase separation by subjecting an initially homogeneous polymer solution inside vesicles to an abrupt osmotic quench. The latter removes water and elevates polymer concentrations in the phase-coexistence regime thereby initiating a segregative phase separation of the polymers. We find that the ensuing relaxation – en route to the new equilibrium – is non-trivially modulated by a dynamic interplay between the coarsening of the evolving droplet phase and the interactive membrane boundary. The early trajectory of droplet coarsening exhibit significant acceleration, but a competing process of membrane-droplet interactions – one in which the membrane boundary is preferentially wetted by one of the incipient phases – dynamically arrests the progression and deforms the membrane. As a result, a novel multi-bud morphology, reminiscent of cellular blebs, decorate the vesicle surface. Furthermore, when the vesicles are composed of phase-separating mixtures of common lipids, the three-dimensional liquid-liquid phase separation within the vesicular interior becomes coupled to the membrane’s compositional degrees of freedom producing microphase-separated membrane textures. This coupling of bulk and surface phase separation processes suggests a new physical principle by which liquid-liquid phase separation inside living cells might be dynamically regulated and materially communicated inside-out to the cellular boundaries.