In the last decade, political debates have progressively shifted to social media. Rhetorical devices employed by online actors and factions that operate in these debating arenas can be captured and analysed to conduct a statistical reading of societal controversies and their argumentation dynamics. In this paper, we propose a five-step methodology, to extract, categorize and explore the latent argumentation structures of online debates. Using Twitter data about a "no-deal" Brexit, we focus on the expected effects in case of materialisation of this event. First, we extract cause-effect claims contained in tweets using RegEx that exploit verbs related to Creation, Destruction and Causation. Second, we categorise extracted "no-deal" effects using a Structural Topic Model estimated on unigrams and bigrams. Third, we select controversial effect topics and explore within-topic argumentation differences between self-declared partisan user factions. We hence type topics using estimated covariate effects on topic propensities, then, using the topics correlation network, we study the topological structure of the debate to identify coherent topical constellations. Finally, we analyse the debate time dynamics and infer lead/follow relations among factions. Results show that the proposed methodology can be employed to perform a statistical rhetorics analysis of debates, and map the architecture of controversies across time. In particular, the "no-deal" Brexit debate is shown to have an assortative argumentation structure heavily characterized by factional constellations of arguments, as well as by polarized narrative frames invoked through verbs related to Creation and Destruction. Our findings highlight the benefits of implementing a systemic approach to the analysis of debates, which allows the unveiling of topical and factional dependencies between arguments employed in online debates.