Recognizing semantic differences across documents is crucial for text generation evaluation and content alignment, especially in cross-lingual settings. However, as a standalone task, it has received little attention. We address this by introducing SwissGov-RSD, the first naturalistic, document-level, cross-lingual dataset for semantic difference recognition. It encompasses a total of 224 multi-parallel documents in English--German, English--French, and English--Italian with token-level difference annotations by human annotators. We evaluate a variety of open-source and closed-source large language models as well as encoder models across different fine-tuning settings on this new benchmark. Our results show that current automatic approaches perform poorly compared to their performance on monolingual, sentence-level, and synthetic benchmarks, revealing a considerable gap for both LLMs and encoder models. We make our code and dataset publicly available.