We propose a framework for active mapping and exploration that leverages Gaussian splatting for constructing dense maps. Further, we develop a GPU-accelerated motion planning algorithm that can exploit the Gaussian map for real-time navigation. The Gaussian map constructed onboard the robot is optimized for both photometric and geometric quality while enabling real-time situational awareness for autonomy. We show through viewpoint selection experiments that our method yields comparable Peak Signal-to-Noise Ratio (PSNR) and similar reconstruction error to state-of-the-art approaches, while being orders of magnitude faster to compute. In closed-loop physics-based simulation and real-world experiments, our algorithm achieves better map quality (at least 0.8dB higher PSNR and more than 16% higher geometric reconstruction accuracy) than maps constructed by a state-of-the-art method, enabling semantic segmentation using off-the-shelf open-set models. Experiment videos and more details can be found on our project page: https://tyuezhan.github.io/RT GuIDE/