Our goal is to develop a modern approach to answering questions via systematic reasoning where answers are supported by human interpretable proof trees grounded in an NL corpus of facts. Such a system would help alleviate the challenges of interpretability and hallucination with modern LMs, and the lack of grounding of current explanation methods (e.g., Chain-of-Thought). This paper proposes a new take on Prolog-based inference engines, where we replace handcrafted rules with a combination of neural language modeling, guided generation, and semiparametric dense retrieval. Our implementation, NELLIE, is the first system to demonstrate fully interpretable, end-to-end grounded QA as entailment tree proof search, going beyond earlier work explaining known-to-be-true facts from text. In experiments, NELLIE outperforms a similar-sized state-of-the-art reasoner while producing knowledge-grounded explanations. We also find NELLIE can exploit both semi-structured and NL text corpora to guide reasoning. Together these suggest a new way to jointly reap the benefits of both modern neural methods and traditional symbolic reasoning.