Liquid-liquid phase separation plays a key role in the assembly of diverse intracellular structures. However, the biophysical principles by which phase separation can be precisely localized within subregions of the cell are still largely unclear, particularly for low-abundance proteins. Here, we introduce an oligomerizing biomimetic system, "Corelets," and utilize its rapid and quantitative light-controlled tunability to map full intracellular phase diagrams, which dictate the concentrations at which phase separation occurs and the transition mechanism, in a protein sequence dependent manner. Surprisingly, both experiments and simulations show that while intracellular concentrations may be insufficient for global phase separation, sequestering protein ligands to slowly diffusing nucleation centers can move the cell into a different region of the phase diagram, resulting in localized phase separation. This diffusive capture mechanism liberates the cell from the constraints of global protein abundance and is likely exploited to pattern condensates associated with diverse biological processes. VIDEO ABSTRACT.