We consider constraint-based methods for causal structure learning, such as the PC algorithm or any PC-derived algorithms whose first step consists in pruning a complete graph to obtain an undirected graph skeleton, which is subsequently oriented. All constraint-based methods perform this first step of removing dispensable edges, iteratively, whenever a separating set and corresponding conditional independence can be found. Yet, constraint-based methods lack robustness over sampling noise and are prone to uncover spurious conditional independences in finite datasets. In particular, there is no guarantee that the separating sets identified during the iterative pruning step remain consistent with the final graph. In this paper, we propose a simple modification of PC and PC-derived algorithms so as to ensure that all separating sets identified to remove dispensable edges are consistent with the final graph,thus enhancing the explainability of constraint-basedmethods. It is achieved by repeating the constraint-based causal structure learning scheme, iteratively, while searching for separating sets that are consistent with the graph obtained at the previous iteration. Ensuring the consistency of separating sets can be done at a limited complexity cost, through the use of block-cut tree decomposition of graph skeletons, and is found to increase their validity in terms of actual d-separation. It also significantly improves the sensitivity of constraint-based methods while retaining good overall structure learning performance. Finally and foremost, ensuring sepset consistency improves the interpretability of constraint-based models for real-life applications.