We derive tractable criteria for the consistency of Bayesian tree reconstruction procedures, which constitute a central class of algorithms for inferring common ancestry among DNA sequence samples in phylogenetics. Our results encompass several Bayesian algorithms in widespread use, such as BEAST, MrBayes, and RevBayes. Unlike essentially all existing asymptotic guarantees for tree reconstruction, we require no discretization or boundedness assumptions on branch lengths. Our results are also very flexible, and easy to adapt to variations of the underlying inference problem. We demonstrate the practicality of our criteria on two examples: a Kingman coalescent prior on rooted, ultrametric trees, and an independence prior on unconstrained binary trees, though we emphasize that our result also applies to non-binary tree models. In both cases, the convergence rate we obtain matches known, frequentist results obtained using stronger boundedness assumptions, up to logarithmic factors.