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In this paper, we present the RISE-SLAM algorithm for performing visual-inertial simultaneous localization and mapping (SLAM), while improving estimation consistency. Specifically, in order to achieve real-time operation, existing approaches often assume previously-estimated states to be perfectly known, which leads to inconsistent estimates. Instead, based on the idea of the Schmidt-Kalman filter, which has processing cost linear in the size of the state vector but quadratic memory requirements, we derive a new consistent approximate method in the information domain, which has linear memory requirements and adjustable (constant to linear) processing cost. In particular, this method, the resource-aware inverse Schmidt estimator (RISE), allows trading estimation accuracy for computational efficiency. Furthermore, and in order to better address the requirements of a SLAM system during an exploration vs. a relocalization phase, we employ different configurations of RISE (in terms of the number and order of states updated) to maximize accuracy while preserving efficiency. Lastly, we evaluate the proposed RISE-SLAM algorithm on publicly-available datasets and demonstrate its superiority, both in terms of accuracy and efficiency, as compared to alternative visual-inertial SLAM systems.