We prove a Carleman estimate in a neighborhood of a multi-interface, that is, near a point where n manifolds intersect, under compatibility assumptions between the Carleman weight, the operators at the multi-interface, and the elliptic operators in the interior and the usual sub-ellipticity condition. The compatibility condition at the multi-interface is a version of the known covering condition for systems in the literature. This Carleman estimate is a generalization of the results obtained in [6, 7]. Applications in control theory for second-order transmission problems are also considered, and provide a generalization of the known results for one-dimensional networks of strings [4, 13, 19, 30] to higher dimensions.