  In this work we apply a local quantum field theory approach in order to
analyse the connection between real-time observables and Euclidean thermal
correlation functions. In particular, using data generated from the functional
renormalisation group in the quark-meson model, we demonstrate that in-medium
effects can be directly extracted from the spatial momentum dependence of the
Euclidean propagators, in contrast to conventional approaches, which rely on
the reconstruction from different Matsubara frequencies. As an application, we
determine the analytic features that arise from the discrete spectral
contribution to the pion correlation function, and calculate the
non-perturbative shear viscosity arising from these states.
