An applied problem facing all areas of data science is harmonizing data sources. Joining data from multiple origins with unmapped and only partially overlapping features is a prerequisite to developing and testing robust, generalizable algorithms, especially in health care. We approach this issue in the common but difficult case of numeric features such as nearly Gaussian and binary features, where unit changes and variable shift make simple matching of univariate summaries unsuccessful. We develop two novel procedures to address this problem. First, we demonstrate multiple methods of "fingerprinting" a feature based on its associations to other features. In the setting of even modest prior information, this allows most shared features to be accurately identified. Second, we demonstrate a deep learning algorithm for translation between databases. Unlike prior approaches, our algorithm takes advantage of discovered mappings while identifying surrogates for unshared features and learning transformations. In synthetic and real-world experiments using two electronic health record databases, our algorithms outperform existing baselines for matching variable sets, while jointly learning to impute unshared or transformed variables.