Trial investigators often have a primary interest in the estimation of the survival curve in a population for which there exists acceptable historical information from which to borrow strength. However, borrowing strength from a historical trial that is non-exchangeable with the current trial can result in biased conclusions. In this article we propose a fully Bayesian semiparametric method for the purpose of attenuating bias and increasing efficiency when jointly modeling time-to-event data from two possibly non-exchangeable sources of information. We illustrate the mechanics of our methods by applying them to a pair of post-market surveillance datasets regarding adverse events in persons on dialysis that had either a bare metal or drug-eluting stent implanted during a cardiac revascularization surgery. We finish with a discussion of the advantages and limitations of this approach to evidence synthesis, as well as directions for future work in this area. The article's Supplementary Materials offer simulations to show our procedure's bias, mean squared error, and coverage probability properties in a variety of settings.