We report on the dynamical formation of exceptional degeneracies in basic correlation functions of nonintegrable one- and two-dimensional systems quenched to the vicinity of a critical point. Remarkably, fine-tuned semimetallic points in the phase diagram of the considered systems are thereby promoted to topologically robust non-Hermitian (NH) nodal phases emerging in the coherent time evolution of a dynamically equilibrating system. Using nonequilibrium Green's function methods within the conserving second Born approximation, we predict observable signatures of these NH nodal phases both in equilibrated spectral functions and in the nonequilibrium dynamics of momentum distribution functions.