Frank’s min–max theorem gives an exact relation between maximum joins and optimal ear decompositions in graphic matroids. In our paper Joins and ear decompositions beyond graphic matroids, we investigate how far this relation extends beyond the graphic setting. The exact equality already fails for cographic matroids, and computing a maximum join is NP-hard even in this class. On the positive side, we prove quantitative comparison bounds for several important classes and, using Seymour’s decomposition theorem, obtain a constant-factor analogue of Frank’s theorem for regular matroids.