Scene understanding is paramount in robotics, self-navigation, augmented reality, and many other fields. To fully accomplish this task, an autonomous agent has to infer the 3D structure of the sensed scene (to know where it looks at) and its content (to know what it sees). To tackle the two tasks, deep neural networks trained to infer semantic segmentation and depth from stereo images are often the preferred choices. Specifically, Semantic Stereo Matching can be tackled by either standalone models trained for the two tasks independently or joint end-to-end architectures. Nonetheless, as proposed so far, both solutions are inefficient because requiring two forward passes in the former case or due to the complexity of a single network in the latter, although jointly tackling both tasks is usually beneficial in terms of accuracy. In this paper, we propose a single compact and lightweight architecture for real-time semantic stereo matching. Our framework relies on coarse-to-fine estimations in a multi-stage fashion, allowing: i) very fast inference even on embedded devices, with marginal drops in accuracy, compared to state-of-the-art networks, ii) trade accuracy for speed, according to the specific application requirements. Experimental results on high-end GPUs as well as on an embedded Jetson TX2 confirm the superiority of semantic stereo matching compared to standalone tasks and highlight the versatility of our framework on any hardware and for any application.