Stereo vision is a key technology for 3D scene reconstruction from image pairs. Most approaches process perspective images from commodity cameras. These images, however, have a very limited field of view and only picture a small portion of the scene. In contrast, omnidirectional images, also known as fisheye images, exhibit a much larger field of view and allow a full 3D scene reconstruction with a small amount of cameras if placed carefully. However, omnidirectional images are strongly distorted which make the 3D reconstruction much more sophisticated. Nowadays, a lot of research is conducted on CNNs for omnidirectional stereo vision. Nevertheless, a significant gap between estimation accuracy and throughput can be observed in the literature. This work aims to bridge this gap by introducing a novel set of transformations, namely OmniGlasses. These are incorporated into the architecture of a fast network, i.e., AnyNet, originally designed for scene reconstruction on perspective images. Our network, Omni-AnyNet, produces accurate omnidirectional distance maps with a mean absolute error of around 13 cm at 36.4 fps and is therefore real-time capable.