This preprint introduces a method for structural phylogenetics using tertiary-interaction characters (3Di) to improve phylogenetic inference beyond the "twilight zone" where sequence similarity is low. The study uses 53 ferritin-like superfamily structures to test whether 3Di characters can be used as standard phylogenetic characters. By combining amino acid sequences, 3Di characters, and custom models, the method produces phylogenies that closely match the structural distances tree of Malik et al. (2020), avoiding long-branch attraction errors. The method is fast, using ultrafast bootstrapping and running in minutes on desktop computers. The 3Di characters are derived from Foldseek, a program that identifies tertiary interactions in protein structures and classifies them into 20 states. The 3Di characters are more conserved than amino acids and can be used in standard phylogenetic analysis. The study shows that 3Di characters can be used in model-based phylogenetic inference, and that combining them with amino acid data improves phylogenetic results. The method is computationally efficient and can be applied to many proteins, making structural phylogenetics more accessible. The study also highlights the potential of 3Di characters as a source of slowly-evolving, phylogenetically informative characters. The results suggest that structural phylogenetics could soon become a routine part of protein phylogenetics, allowing the re-exploration of many fundamental phylogenetic problems. The study also discusses the limitations of current methods and the potential of 3Di characters in phylogenetic analysis. The results indicate that 3Di characters are robust to structural variations and can be used in phylogenetic inference. The study concludes that 3Di characters are a promising tool for structural phylogenetics and that further research is needed to fully understand their potential.This preprint introduces a method for structural phylogenetics using tertiary-interaction characters (3Di) to improve phylogenetic inference beyond the "twilight zone" where sequence similarity is low. The study uses 53 ferritin-like superfamily structures to test whether 3Di characters can be used as standard phylogenetic characters. By combining amino acid sequences, 3Di characters, and custom models, the method produces phylogenies that closely match the structural distances tree of Malik et al. (2020), avoiding long-branch attraction errors. The method is fast, using ultrafast bootstrapping and running in minutes on desktop computers. The 3Di characters are derived from Foldseek, a program that identifies tertiary interactions in protein structures and classifies them into 20 states. The 3Di characters are more conserved than amino acids and can be used in standard phylogenetic analysis. The study shows that 3Di characters can be used in model-based phylogenetic inference, and that combining them with amino acid data improves phylogenetic results. The method is computationally efficient and can be applied to many proteins, making structural phylogenetics more accessible. The study also highlights the potential of 3Di characters as a source of slowly-evolving, phylogenetically informative characters. The results suggest that structural phylogenetics could soon become a routine part of protein phylogenetics, allowing the re-exploration of many fundamental phylogenetic problems. The study also discusses the limitations of current methods and the potential of 3Di characters in phylogenetic analysis. The results indicate that 3Di characters are robust to structural variations and can be used in phylogenetic inference. The study concludes that 3Di characters are a promising tool for structural phylogenetics and that further research is needed to fully understand their potential.