The article presents the results of further development of the previously proposed by the authors' new asymptotic theory of thin multilayer anisotropic shells. The theory is constructed on the equations of the general three-dimensional theory of elasticity by introducing small geometric parameter asymptotic expansions without any hypotheses on stresses and displacements distribution over the thickness. The case of monoclinic layers having at the most 13 independent elastic constants is considered. An algorithm for obtaining explicit analytic formulas for the calculation of the complete stress tensor component distribution over the shell is proposed. The algorithm is based on solving specific local problems of the first, second and third approximations. It allows obtaining expressions for all six components of the stress tensor in a compact closed form, as a function of strain, curvature of the middle surface of the shell, as well as their derivatives with respect to the longitudinal coordinates. These formulas allow calculating all stress tensor components in the shell without additional tasks, using only the solutions of the averaged problem of shell theory.