Line drawing is an important and concise method to depict the shape of an object. Stereo line drawing, a combination of line drawing and stereo rendering, not only efficiently conveys shape but also provides users with a visual experience of a stereoscopic 3D world. Contours are the most important lines to draw. However, contours must be rendered consistently for two eyes because of their view-dependent nature; otherwise, they cause binocular rivalry and viewing discomfort. This paper proposes a novel solution to draw stereo-consistent contours in real time. First, we extend the concept of epipolar-slidability and derive a new criterion to check epipolar-slidability by the monotonicity of the trajectory of the viewpoints of contour points. Then, we design an algorithm to test the epipolar-slidability of contours by conducting an image space search rather than sampling multiple viewpoints. Results show that the proposed method has a much lower cost than that of previous works, therefore enables the real-time rendering and editing of stereo-consistent contours for users, such as changing camera viewpoints, editing object geometry, tweaking parameters to show contours with different details, etc.