Matched-field processing (MFP) and matched-mode processing (MMP) are two popular techniques for passively localizing an underwater acoustic emitter in range and depth. One major drawback of these techniques has been their sensitivity to uncertainty concerning the acoustic environment. Several methods of addressing this phenomenon have been proposed in the literature, with varying degrees of success. Achieving high-quality location estimates remains a problem except in simple range-independent experiments or numerical simulations. In this study, an approach is demonstrated for robust, accurate emitter localization in a highly range-dependent real environment using MMP. The main factors contributing to successful localization are: (1) use of the high-resolution multiple signal classification (MUSIC) algorithm, which performs well even when only a few robust modes can be obtained by mode filtering; and (2) use of an acoustic propagation model incorporating mode coupling, which is able to generate accurate replica fields in a strongly range-dependent environment. The use of a cumulant-based MUSIC algorithm for reducing noise effects is also investigated.