Understanding how extreme price movements propagate across financial and energy markets is critical for risk management and regulatory design in the EU Emissions Trading System (EU ETS). We apply Hüsler-Reiss graphical models of extremes to a system of 20 daily variables centred on EU allowances futures across Phases 3 and 4 of the EU ETS (2013--2025), with a Gaussian graphical model as the average-dependence baseline. The tail networks are structurally distinct from the average dependence network: substantially denser, organized around different central nodes, and governed by within-sector homophily that binds sector boundaries more tightly than at the average-dependence level. EU allowances futures are peripheral in the standard graphical model but achieve the highest centrality in the tail networks, while equity indices and major FX pairs follow the opposite trajectory. Exponential random graph models confirm equity and FX peripherality in tail networks across all sample periods and identify triadic closure during market downturns as a Phase~3 phenomenon that vanishes in Phase~4. The phase transition restructures the tail network without thinning it: average dependence contracts sharply while tail dependence persists, and crash contagion shifts from clustered to diffuse propagation. These findings have direct implications for hedge construction by compliance entities, stress-test calibration by regulators, and the design of systemic-risk monitoring tools for EU ETS markets.

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