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Monolingual word alignment is important for studying fine-grained editing operations (i.e., deletion, addition, and substitution) in text-to-text generation tasks, such as paraphrase generation, text simplification, neutralizing biased language, etc. In this paper, we present a novel neural semi-Markov CRF alignment model, which unifies word and phrase alignments through variable-length spans. We also create a new benchmark with human annotations that cover four different text genres to evaluate monolingual word alignment models in more realistic settings. Experimental results show that our proposed model outperforms all previous approaches for monolingual word alignment as well as a competitive QA-based baseline, which was previously only applied to bilingual data. Our model demonstrates good generalizability to three out-of-domain datasets and shows great utility in two downstream applications: automatic text simplification and sentence pair classification tasks.