Biochemical networks interconnect, grow and evolve to express new properties as different chemical pathways are selected during a continuous cycle of energy consumption and transformation. In contrast, synthetic systems that push away from equilibrium usually return to the same self-assembled state, often generating waste that limits system recyclability and prevents the formation of adaptable networks. Here we show that annealing by slow proton dissipation selects for otherwise inaccessible morphologies of fibres built from DNA and cyanuric acid. Using single-molecule fluorescence microscopy, we observe that proton dissipation influences the growth mechanism of supramolecular polymerization, healing gaps within fibres and converting highly branched, interwoven networks into nanocable superstructures. Just as the growth kinetics of natural fibres determine their structural attributes to modulate function, our system of photoacid-enabled depolymerization and repolymerization selects for healed materials to yield organized, robust fibres. Our method provides a chemical route for error-checking, distinct from thermal annealing, that improves the morphologies and properties of supramolecular materials using out-of-equilibrium systems.