Vertex-reinforced jump processes are stochastic processes in continuous time that prefer to jump to sites that have accumlated a large local time. Sabot and Tarrès showed interesting connections between vertex-reinforced jump processes and a supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer in a completely different context. In the talk, I will present an extension of Zirnbauer's model and show how it arises naturally as a weak joint limit of a time-changed version of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees. The talk is based on joint work with Franz Merkl and Pierre Tarrès.