We consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the Laplacian in the exterior, connected through standard transmission conditions. We establish a DPG scheme coupled with Galerkin boundary elements for its discretization, and prove its robustness for the field variables in so-called balanced norms. Our coupling scheme is the one from Fuhrer et al., (2016), adapted to the singularly perturbed case by using the scheme from Heuer and Karkulik (2015). Essential feature of our method is that optimal test functions have to be computed only locally. We report on various numerical experiments in two dimensions. (C) 2016 Elsevier Ltd. All rights reserved.