Some applications require the interactive resolution of a constraint problem by a human user. In such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inverse consistency. Existing systems simulate this either by maintaining arc consistency after each assignment performed by the user or by compiling offline the problem as a multi-valued decision diagram. In this article, we define several questions related to global inverse consistency and analyze their complexity. Despite their theoretical intractability, we propose several algorithms for enforcing and restoring global inverse consistency and we show that the best version is efficient enough to be used in an interactive setting on several configuration and design problems.