In this letter, we present a neural field-based real-time monocular mapping framework for accurate and dense Simultaneous Localization and Mapping (SLAM). Recent neural mapping frameworks show promising results, but rely on RGB-D or pose inputs, or cannot run in real-time. To address these limitations, our approach integrates dense-SLAM with neural implicit fields. Specifically, our dense SLAM approach runs parallel tracking and global optimization, while a neural field-based map is constructed incrementally based on the latest SLAM estimates. For the efficient construction of neural fields, we employ multi-resolution grid encoding and signed distance function (SDF) representation. This allows us to keep the map always up-to-date and adapt instantly to global updates via loop closing. For global consistency, we propose an efficient Sim(3)-based pose graph bundle adjustment (PGBA) approach to run online loop closing and mitigate the pose and scale drift. To enhance depth accuracy further, we incorporate learned monocular depth priors. We propose a novel joint depth and scale adjustment (JDSA) module to solve the scale ambiguity inherent in depth priors. Extensive evaluations across synthetic and real-world datasets validate that our approach outperforms existing methods in accuracy and map completeness while preserving real-time performance.