We study the equilibrium and structural properties of mixtures of like-charged colloids and nonadsorbing flexible polyelectrolytes using liquid state integral equations. The polymer chains in the semidilute regime are treated within the polymer reference interaction site model. The monomers and the colloids interact via screened Coulomb potentials (Debye-Huckel level). The electrostatic interactions between the different macromolecules inside the solution are responsible for complex local structures and rich phase diagrams, which depend on the charges and on the Debye screening length. The pair correlations between colloids are interpreted in terms of the effective, polymer-averaged, colloid-colloid pair potential. A first-order picture indicates that the Coulombic depletion attraction due to the osmotic pressure of the semidilute network against the colloidal particles dominates the direct repulsion at high enough monomer concentration and induces the liquid-liquid phase separation. Higher-order effects involving correlations inside the polyelectrolyte system alter this simple picture.