The dynamics of a fluid surface filled with high-amplitude ripples were studied with a technique (diffusing light photography) that resolves the height at all locations instantaneously. Even when nonlinearities are strong enough to generate a (Kolmogorov) cascade from long wavelength (where energy is input) to shorter wavelength, the resulting turbulent state contains large coherent spatial structures. The appearance of these structures in a thermal equilibrium state (with the same average energy) would be highly improbable.