Ontology alignment agreement aims at overcoming the problem that arises when different parties need to conciliate their conflicting views on ontology alignments. Argumentation has been applied as a way for supporting the creation and exchange of arguments, followed by the reasoning on their acceptability. Here we use arguments as positions that support or reject correspondences. Applying only argumentation to select correspondences may lead to alignments which relates ontologies in an inconsistent way. In order to address this problem, we define maximal consistent sub-consolidations which generate consistent and argumentation-grounded alignments. We propose a strategy for computing them involving both argumentation and logical inconsistency detection. It removes correspondences that introduce inconsistencies into the resulting alignment and allows for maintaining the consistency within an argumentation system. We present experiments comparing the different approaches. The (partial) experiments suggest that applying consistency checking and argumentation independently significantly improves results, while using them together does not bring so much. The features of consistency checking and argumentation leading to this result are analysed.