Liquid-liquid phase separation (LLPS) mediates formation of membraneless condensates such as those associated with RNA processing, but the rules that dictate their assembly, substructure, and coexistence with other liquid-like compartments remain elusive. Here, we address the biophysical mechanism of this multiphase organization using quantitative reconstitution of cytoplasmic stress granules (SGs) with attached P-bodies in human cells. Protein-interaction networks can be viewed as interconnected complexes (nodes) of RNA-binding domains (RBDs), whose integrated RNA-binding capacity determines whether LLPS occurs upon RNA influx. Surprisingly, both RBD-RNA specificity and disordered segments of key proteins are non-essential, but modulate multiphase condensation. Instead, stoichiometry-dependent competition between protein networks for connecting nodes determines SG and P-body composition and miscibility, while competitive binding of unconnected proteins disengages networks and prevents LLPS. Inspired by patchy colloid theory, we propose a general framework by which competing networks give rise to compositionally specific and tunable condensates, while relative linkage between nodes underlies multiphase organization.