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We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on R n . In the spirit of domain decomposition, we partition R n = Ω ∪ Γ ∪ Ω + , Ω a bounded Lipschitz polyhedron, Γ : = ∂ Ω , and Ω + unbounded. In Ω , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In Ω + , we rely on a meshless Trefftz-Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across Γ . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results., (© The Author(s) 2022.)