Narratives determine our worldviews, but need evidence to be believable. For scientific narratives, hard evidence usually is provided by specially curated research data and experiments within each publication. The aim is to strengthen the narratives' overall plausibility: the less plausible a narrative is, the more it is bound to be challenged. As recent advances in NLP enabled the scientific community to tackle important problems in fact-checking and more profound semantic interpretations of structured data, there is now also hope to unlock the rich narrative knowledge that such data sets can offer. Yet, current strategies to extract such narrative knowledge still heavily rely on exhaustive bottom-up analysis to cast insights from data into a human-understandable form. In this paper, we take a novel integration-based approach to design a system that reduces the task of finding narrative evidence to applying a sequence of simpler top-down matching tasks. Our BiND system builds upon an expressive definition of structured narratives. It uses them as templates for schema and instance matching against web tables, thereby computing flexible bindings between narratives and data. By combining structured narratives with a carefully chosen selection of statistical metrics to assess the inherent relationships between different attributes of the matched data, our system allows us to reliably identify the most plausible witnesses for a given narrative. We demonstrate the applicability of our system in the real world on the vast open data repository of the World Health Organization (WHO).