Identification of nonminimum phase systems with finite impulse response is addressed in the bispectrum domain. A bispectrum-based phase retrieval algorithm is modified to handle the phase wrapping problem, and is extended to log-magnitude reconstruction. Both linear equation based estimators are then combined, to form an integrated, nonparametric system identification method. Weighted forms of the above estimators are developed, which are asymptotically minimum-variance in the class of weighted least-squares estimators. Asymptotic variance expressions are derived for both the weighted and the unweighted forms. Theory and simulations illustrate that the new approaches can identify nonminimum phase MA models, using output-data that may be corrupted by additive Gaussian noise of unknown covariance. Due to their nonparametric nature, the proposed algorithms outperform existing linear equation cumulant-based modeling methods, in the case of model order mismatch.