This paper addresses the problem of matched-field passive localization of broadband underwater acoustic sources with a short vertical array. In previous work, incoherent averaging of narrow-band Bartlett ambiguity surfaces has been proposed to improve the robustness of matched-field processing (MFP) to environmental uncertainties. While computationally efficient, the effectiveness of this approach is dependent on the type of environmental mismatch present. In this paper, to provide robust source localization performance for short arrays, an alternative wideband MFP method is proposed which utilizes the fourth-order statistics of the data. The method is 'coherent' in that it exploits cross-frequency dependencies in the random acoustic channel response. The proposed coherent wideband minimum variance method with environmental perturbation constraints (MV-EPC) consists of minimizing the squared output power of a beamformer subject to constraints defined across the signal band. The constraints are designed to provide robustness over an ensemble of random environmental realizations. Monte Carlo simulation results using a canonical shallow-water scenario indicate that the coherent wideband MV-EPC method yields improved probability of correct localization performance versus incoherent averaging of narrow-band Bartlett ambiguity functions with a short array above a threshold signal-to-noise ratio.