An explicit formula is obtained for single-phase bounded elliptic solutions of the Manakov system of integrable coupled nonlinear Schrödinger equations in terms of the Weierstrass sigma function with a real quasiperiod. The parametrization is effective in the sense that the reality conditions are completely characterized for each of the three possible couplings: focusing–focusing, defocusing–defocusing and focusing–defocusing. The Manakov soliton is recovered in the soliton limit and the small-wave-modulation limit is shown to satisfy the linearized dispersion relation of planewave solutions.