<p>Ray-theory computations of shear (S) waves in inhomogeneous anisotropic media collapse in cases when the two S waves, existing in such media, propagate close to each other. This happens in vicinities of shear-wave singularities or, generally, in weakly anisotropic media. Described problems can be removed by the use of the coupling ray theory. Along a ray, called common S-wave ray, one has to solve repeatedly a system of two frequency-dependent ordinary differential equations. Solutions are sought for frequencies covering the spectrum of the used signal. This complicates computations considerably. In this paper, we use recently proposed approximation, called prevailing-frequency approximation, and apply it to the S-wave coupling ray theory based on the weak-anisotropy approximation. Use of the prevailing-frequency approximation removes the requirement of repeated solution of ordinary differential equations. Instead of it, one has to solve just four ordinary differential equations along a common S-wave ray. It makes the procedure of computing S-wave fields similar to the procedure used for separate S waves or for P waves. In this paper, we derive basic formulae and discuss their properties. We test the derived formulae in specific situations (weak anisotropy, vicinity of a singularity, etc.) in the companion paper. </p>