Note: Funding Information: We thank N. Lartillot for assistance in investigating the posterior predictive test reproducibility issue, J. Viklund for updating the alignment_pruner.pl script, and A. Roger and K. Williamson for helpful insights and discussions. Phylobayes-MPI computations were performed on resources provided by the PDC Center for High-Performance Computing (PDC) under Project SNIC 2020/5-473. This work was supported by grants from the European Research Council Consolidator (grant 817834), the Dutch Research Council (NWO; VICI grant VI.C.192.016) to T.J.G.E. and the Swedish Research Council (VR International PostDoc grant 2018-06727) to J.M.