Many biological materials are composite structures, interpenetrating networks of different types of fibers. The composite nature of such networks leads to superior mechanical properties, but the origin of this mechanical synergism is still poorly understood. Here we study soft composite networks, made by mixing two self-assembling fiber-forming components. We find that the elastic moduli of the composite networks significantly exceed the sum of the moduli of the two individual networks. This mechanical enhancement is in agreement with recent simulations, where it was attributed to a suppression of non-affine deformation modes in the most rigid fiber network due to the reaction forces in the softer network. The increase in affinity also causes a loss of strain hardening and an increase in the critical stress and stain at which the network fails.