  Imaging with hard x-rays is an invaluable tool in medicine, biology,
materials science, and cultural heritage. Propagation-based x-ray
phase-contrast imaging and tomography have been mostly used to resolve
micrometer-scale structures inside weakly absorbing objects as well as inside
dense specimens. Indirect x-ray detection has been the key technology to
achieve up to sub-micrometer spatial resolutions, albeit inefficiently and
hence at the expense of increased radiation dose to the specimen. A promising
approach to low-dose imaging and high spatial resolution even at high x-ray
energies is ghost imaging, which could use single-pixel, yet efficient direct
x-ray detectors made of high-density materials. However, phase contrast has not
yet been realised with x-ray ghost imaging. We present an approach which
exploits both the advantages of x-ray ghost imaging and the high sensitivity of
phase-contrast imaging. In comparison with existing techniques, our method is
efficient and achieves high-fidelity x-ray ghost images with phase contrast,
accurate density resolution and dramatically higher spatial resolution. The
method is scalable to practical tomography with large fields of view,
micrometer spatial resolution, and with high-energy x-rays above 100 keV. It is
also applicable to other phase-sensitive imaging techniques and with other
probes such as neutrons, alpha rays, and muons, for which high spatial
resolution detectors are limited or even not available.
