We report on the formation of induced dark solitary pulses in a net anomalous dispersion cavity fiber laser. In a weak birefringence cavity fiber laser simultaneous laser oscillation along the two orthogonal polarization directions of the cavity could be achieved. Under suitable conditions bright cavity solitons could be formed along one polarization direction while CW emission occurs along the orthogonal polarization direction. In a previous paper we have shown that under incoherent polarization coupling a bright soliton always induces a broad dark pulse on the CW beam. In the paper we further show that under coherent polarization coupling a bright soliton could further induce either a weak bright or a dark solitary pulse on the bottom of the broad dark pulse. Numerical simulations have also well reproduced the experimental observations, and further show whether a weak dark or bright solitary pulse is induced is determined by the presence or absence of a phase jump in the induced pulse.