We consider the relativistic multi-mode E-3 x (t(2) + e) Jahn-Teller effect for tetrahedral atomic complexes in a triplet electronic state. Our analysis is based on the symmetry properties of the electronic Hamiltonian and includes spin-orbit coupling terms. The Taylor series expansion of the electronic Hamiltonian in terms of normal modes is restricted to second order in the electrostatic contributions and to first order in the spin-orbitspin-orbit contributions. The main result of this work is a 6 x 6 vibronic matrix which depends on the t(2) and e vibrational normal modes and is specified by six electrostatic and three spin-orbit coupling constants. (C) 2018 Elsevier B.V. All rights reserved.