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Demixing of binary liquids is a ubiquitous transition, which is explained using a well-established thermodynamic formalism that requires equality of intensive thermodynamics parameters across the phase boundaries. Demixing transitions also occur when binary fluid mixtures are driven away from equilibrium, for example, by external shear flow. Predicting demixing transition under non-equilibrium non-potential conditions remains, however, a challenge. We drive liquid-liquid phase separation (LLPS) of attractive DNA nanostar molecules away from equilibrium using an internally driven microtubule-based active fluid. Activity lowers the critical temperature and narrows the coexistence concentrations, but only when there are mechanical bonds between the liquid droplets and the reconfiguring active fluid. Similar behaviors are observed in numerical simulations, suggesting that activity suppression of liquid-liquid phase separation is a generic feature of active LLPS. Our work describes a platform for building soft active materials with feedback control while also providing insight into cell biology, where phase separation emerged as a ubiquitous self-organizational principle.