We present the first fully trainable semantic parser for English, German, Italian, and Dutch discourse representation structures (DRSs) that is competitive in accuracy with recent sequence-to-sequence models and at the same time compositional in the sense that the output maps each token to one of a finite set of meaning fragments, and the meaning of the utterance is a function of the meanings of its parts. We argue that this property makes the system more transparent and more useful for human-in-the-loop annotation. We achieve this simply by casting DRS parsing as a sequence labeling task, where tokens are labeled with both fragments (lists of abstracted clauses with relative referent indices indicating unification) and symbols like word senses or names. We give a comprehensive error analysis that highlights areas for future work.