Accurately perceiving an object's pose and shape is essential for precise grasping and manipulation. Compared to common vision-based methods, tactile sensing offers advantages in precision and immunity to occlusion when tracking and reconstructing objects in contact. This makes it particularly valuable for in-hand and other high-precision manipulation tasks. In this work, we present GelSLAM, a real-time 3D SLAM system that relies solely on tactile sensing to estimate object pose over long periods and reconstruct object shapes with high fidelity. Unlike traditional point cloud-based approaches, GelSLAM uses tactile-derived surface normals and curvatures for robust tracking and loop closure. It can track object motion in real time with low error and minimal drift, and reconstruct shapes with submillimeter accuracy, even for low-texture objects such as wooden tools. GelSLAM extends tactile sensing beyond local contact to enable global, long-horizon spatial perception, and we believe it will serve as a foundation for many precise manipulation tasks involving interaction with objects in hand. The video demo, code, and dataset are available at https://joehjhuang.github.io/gelslam.