Single-distance phase retrieval at large phase shifts. For coherent X-ray imaging of pure phase objects we study the reliability of linear relations in phase-retrieval algorithms based on a single intensity map after free-space propagation. For large phase changes and/or large propagation distances we propose two venues of working beyond linearity: Projection onto an effective, linear and local model in Fourier space and expansion of intensity contrast in powers of object-detector distance. We apply both algorithms successfully to simulated data.