We present an approximate procedure for computing coupled S waves in inhomogeneous weakly anisotropic media. The procedure can be used to compute S waves propagating in smooth inhomogeneous isotropic or anisotropic media. In anisotropic media, it can describe behaviour of coupled as well as decoupled S waves. Basic part of the procedure is an approximate computation of the common S-wave ray, a trajectory, along which properties of both S waves propagating in anisotropic media are computed. For computation of the common ray and of the approximate geometrical spreading along it, we use the first-order ray tracing and dynamic ray tracing concept, developed for computations of P waves in inhomogeneous weakly anisotropic media. The amplitude coefficients of the coupled S waves are computed by solving a coupled system of ordinary differential equations - the coupling equations - along the common S-wave ray. The performance of the procedure is illustrated on several models of varying strength of anisotropy. Its accuracy is tested by comparing its results with results of the standard ray theory.