In a data exchange setting with target constraints, it is often the case that a given source instance has no solutions. Intuitively, this happens when data sources contain inconsistent or conflicting information that is exposed by the target constraints at hand. In such cases, the semantics of target queries trivialize, because the certain answers of every target query over the given source instance evaluate to “true”. The aim of this paper is to introduce and explore a new framework that gives meaningful semantics in such cases using the notion of exchange-repairs. Informally, an exchange-repair of a source instance is another source instance that differs minimally from the first, but has a solution. In turn, exchange-repairs give rise to a natural notion of exchange-repair certain answers (in short, XR-certain answers) for target queries in the context of data exchange with target constraints. After exploring the structural properties of exchange-repairs, we focus on the problem of computing the XR-certain answers of conjunctive queries. We show that for schema mappings specified by source-to-target GAV dependencies and target equality-generating dependencies (egds), the XR-certain answers of a target conjunctive query can be rewritten as the consistent answers (in the sense of standard database repairs) of a union of conjunctive queries over the source schema with respect to a set of egds over the source schema, thus making it possible to use a consistent query answering system to compute XR-certain answers in data exchange. In contrast, we show that this type of rewriting is neither possible for schema mappings specified by source-to-target LAV dependencies and target egds, nor for schema mappings specified by both source-to-target and target GAV dependencies. We then examine the general case of schema mappings specified by source-to-target GLAV constraints, a weakly acyclic set of target tgds and a set of target egds. The main result asserts that, for such settings, the XR-certain answers of conjunctive queries can be rewritten as the certain answers of a union of conjunctive queries with respect to the stable models of a disjunctive logic program over a suitable expansion of the source schema.