We apply a Bayesian structure learning approach to study interactions between global teleconnection modes, illustrating its use as a framework for developing process-based diagnostics with which to evaluate climate models. Homogeneous dynamic Bayesian network models are constructed for time series of empirical indices diagnosing the activity of major tropical, Northern and Southern Hemisphere modes in the NCEP/NCAR and JRA-55 reanalyses. The resulting probabilistic graphical models are comparable to Granger causal analyses that have recently been advocated. Reversible jump Markov Chain Monte Carlo is employed to provide a quantification of the uncertainty associated with the selection of a single network structure. In general, the models fitted from the NCEP/NCAR reanalysis and the JRA-55 reanalysis are found to exhibit broad agreement in terms of associations for which there is high posterior confidence. Differences between the two reanalyses are found that involve modes for which known biases are present or that may be attributed to seasonal effects, as well as for features that, while present in point estimates, have low overall posterior mass. We argue that the ability to incorporate such measures of confidence in structural features is a significant advantage provided by the Bayesian approach, as point estimates alone may understate the relevant uncertainties and yield less informative measures of differences between products when network-based approaches are used for model evaluation.