In this paper, we introduce MIMII DUE, a new dataset for malfunctioning industrial machine investigation and inspection with domain shifts due to changes in operational and environmental conditions. Conventional methods for anomalous sound detection face practical challenges because the distribution of features changes between the training and operational phases (called domain shift) due to various real-world factors. To check the robustness against domain shifts, we need a dataset that actually includes domain shifts, but such a dataset does not exist so far. The new dataset we created consists of the normal and abnormal operating sounds of five different types of industrial machines under two different operational/environmental conditions (source domain and target domain) independent of normal/abnormal, with domain shifts occurring between the two domains. Experimental results showed significant performance differences between the source and target domains, indicating that the dataset contains the domain shifts. These findings demonstrate that the dataset will be helpful for checking the robustness against domain shifts. The dataset is a subset of the dataset for DCASE 2021 Challenge Task 2 and freely available for download at https://zenodo.org/record/4740355