When intersecting non-matching three-dimensional lattices, one needs to calculate the intersections of tetrahedra. The authors’ previously published two-dimensional triangle–triangle intersection algorithm suggests a novel approach in three dimensions based on parsimony. The algorithm presented here expands on this two-dimensional algorithm and introduces new strategies necessitated by the increase in dimension. An extensive proof is given for the consistency of the algorithm. Thus, the algorithm is shown to be robust to numerical error arising from floating-point arithmetic. Example problems demonstrate its use and effectiveness.