Membrane-less organelles, the liquid droplets formed via liquid-liquid phase separation (LLPS) of biomolecules in cells, act to organize intracellular components into multiple compartments. As a model for this process, and as a potential vehicle for in vitro exploitation of its properties, we explore here a synthetic multiphase LLPS system consisting of a mixture of self-assembled DNA particles. The particles, termed "DNA nanostars" (NSs), consist of four double-stranded DNA arms that each terminate in a single-stranded overhang. NSs condense into droplets due to overhang hybridization. Using two types of NSs with orthogonal overhangs enables the creation of two types of immiscible DNA droplets. Adhesion between the droplets can be tuned by the addition of "cross-linker NSs" that have two overhang sequences of each type. We find that increasing the amount of the cross-linker NSs decreases the droplet/droplet surface tension until a microemulsion transition occurs. Controlled droplet adhesion can also be achieved, without cross-linkers, using overhangs that can weakly hybridize. Finally, we show that solutes can be specifically targeted to the DNA phases by labeling them with appropriate sticky-ends. Overall, our findings demonstrate the ability to create a multiphase LLPS system, and to control its mesoscale configuration, via sequence design of the component molecules.