Collaborative Simultaneous Localization and Mapping (C-SLAM) is a fundamental capability for multi-robot teams as it enables downstream tasks like planning and navigation. However, existing C-SLAM back-end algorithms that are required to solve this problem struggle to address the practical realities of real-world deployments (i.e. communication limitations, outlier measurements, and online operation). In this paper we propose Robust Incremental Manifold Edge-based Separable ADMM (riMESA) -- a robust, incremental, and distributed C-SLAM back-end that is resilient to outliers, reliable in the face of limited communication, and can compute accurate state estimates for a multi-robot team in real-time. Through the development of riMESA, we, more broadly, make an argument for the use of Consensus Alternating Direction Method of Multipliers as a theoretical foundation for distributed optimization tasks in robotics like C-SLAM due to its flexibility, accuracy, and fast convergence. We conclude this work with an in-depth evaluation of riMESA on a variety of C-SLAM problem scenarios and communication network conditions using both synthetic and real-world C-SLAM data. These experiments demonstrate that riMESA is able to generalize across conditions, produce accurate state estimates, operate in real-time, and outperform the accuracy of prior works by a factor >7x on real-world datasets.

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