A common assumption for argumentation-based dialogues is that any argument exchanged is complete, in the sense that its premises entail its claim. However, in real world dialogues, agents commonly exchange enthymemes --- arguments with incomplete logical structure. This paper formalises the dialogical exchange of enthymemes that are missing some constituent elements, such that it is not possible to directly entail the claim of the intended argument from the premises of the enthymeme exchanged. This can lead to misunderstandings between agents; we provide a rich set of locutions for identifying and resolving such misunderstandings, and a protocol that governs the use of these. We show that, under certain conditions, the status of moves made during a dialogue conforming to our system corresponds with the status of arguments in the Dung argument framework instantiated by the contents of the moves made at that stage in the dialogue. This is significant since it ensures that the use of enthyememes does not prevent the agents from reaching the appropriate decision according to the information they have shared.