This article presents a coupling approach for the approximation of iterated stochastic integrals of length three. The generation of such integrals is the central problem of higher-order pathwise approximations for SDEs, which still lacks a satisfactory answer due to the restriction of dimensionality and computational load. Here we start from the Fourier representation of the triple stochastic integral and investigate the global behaviour of the joint density of the representation. Finally in the main result we give a coupling in the quadratic Vaserstein distance.