For a class of generalized hidden Markov models (X_t,Y_t)_t∈ℤ we analyze the limiting behavior of the (suitably scaled) unobservable part (Y_t)_t∈ℤ under an observable extreme event |X₀|>x, as x→∞. We discuss sufficient conditions for the existence of this limit and characterize its special structure. Our approach gives rise to an efficient and flexible algorithm for the Monte Carlo evaluation of extremal characteristics (such as the extremal index) of the observable process. Further, our setup allows to evaluate extremal measures which depend on the extremal behavior of X₋₁,X₋₂,…, i.e. before X₀. An application to financial asset returns is given by the asymmetric GARCH(1,1) model whose extremal behavior has not been considered before. Our results complement the findings of Segers on the tail chains of single time series (Segers 2007).