For high-energy X-ray phase tomography, an inverse Talbot-Lau interferometer using a structured-anode X-ray source has been constructed. The structured anode has a tungsten line-and-space pattern whose period is 3 μm. Thanks to the inverse geometry, a thick amplitude grating with a moderate aspect ratio is available for G2, resulting in a design energy of 82 keV. After confirming the operation of the source as expected, high-energy X-ray phase tomography was applied to a dry battery, which shows little transmission around 30 keV. Structural change in the negative electrode is depicted between tomograms reconstructed from scattering (dark-field) images of a fresh dry battery and identical fully-discharged one. From the viewpoint of practical application to nondestructive testing and medicine, exploring high-energy X-ray phase imaging is important. This result suggests that the use of the structured-anode X-ray source is effective for this purpose thus avoiding the challenge of developing extremely high-aspect-ratio gratings. If the period of the structured anode is reduced further, it will also be possible to remove G2.