Given a collection of hypersurfaces in a complex toric variety, subject to a "cleanness" assumption on their intersection, Li Li and others have constructed wonderful compactifications of their complement. We give a combinatorial, or if you like tropical, description of the Chow ring thereof. This generalises the matroid Chow ring of Feichtner--Yuzvinsky (of recent fame in work of Huh et al.) and reveals the geometry underpinning the description given by de Concini and Gaiffi for subtori.

Joint work in progress with Omid Amini, Emanuele Delucchi, Diane Maclagan, Nolan Schock.