We address a practical scenario of anomaly detection for industrial sound data, where the sound of a target machine is corrupted by background noise and interference from neighboring machines. Overcoming this challenge is difficult since the interference is often virtually indistinguishable from the target machine without additional information. To address the issue, we propose SSAD, a framework of source separation (SS) followed by anomaly detection (AD), which leverages machine activity information, often readily available in practical settings. SSAD consists of two components: (i) activity-informed SS, enabling effective source separation even given interference with similar timbre, and (ii) two-step masking, robustifying anomaly detection by emphasizing anomalies aligned with the machine activity. Our experiments demonstrate that SSAD achieves comparable accuracy to a baseline with full access to clean signals, while SSAD is provided only a corrupted signal and activity information. In addition, thanks to the activity-informed SS and AD with the two-step masking, SSAD outperforms standard approaches, particularly in cases with interference. It highlights the practical efficacy of SSAD in addressing the complexities of anomaly detection in industrial sound data.