Spin-orbit coupling in two-dimensional systems is usually characterized by Rashba and Dresselhaus spin-orbit coupling (SOC) linear in the wave vector. However, there is a growing class of materials which instead support dominant SOC cubic in the wave vector (cSOC), while their superconducting properties remain unexplored. By focusing on Josephson junctions in Zeeman field with superconductors separated by a normal cSOC region, we reveal a strongly anharmonic current-phase relation and complex spin structure. An experimental cSOC tunability enables both tunable anomalous phase shift and supercurrent, which flows even at the zero-phase difference in the junction. A fingerprint of cSOC in Josephson junctions is the $f$-wave spin-triplet superconducting correlations, important for superconducting spintronics and supporting Majorana bound states.

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• Electronic structure
• Josephson effect
• Spin-orbit coupling