SNS providers are known to carry out the recompression and resizing of uploaded images, but most conventional methods for detecting fake images/tampered images are not robust enough against such operations. In this paper, we propose a novel method for detecting fake images, including distortion caused by image operations such as image compression and resizing. We select a robust hashing method, which retrieves images similar to a query image, for fake-image/tampered-image detection, and hash values extracted from both reference and query images are used to robustly detect fake-images for the first time. If there is an original hash code from a reference image for comparison, the proposed method can more robustly detect fake images than conventional methods. One of the practical applications of this method is to monitor images, including synthetic ones sold by a company. In experiments, the proposed fake-image detection is demonstrated to outperform state-of-the-art methods under the use of various datasets including fake images generated with GANs.