Reconstructing the 3D geometry of the surgical site and detecting instruments within it are important tasks for surgical navigation systems and robotic surgery automation. Traditional approaches treat each problem in isolation and do not account for the intrinsic relationship between segmentation and stereo matching. In this paper, we present a learning-based framework that jointly estimates disparity and binary tool segmentation masks. The core component of our architecture is a shared feature encoder which allows strong interaction between the aforementioned tasks. Experimentally, we train two variants of our network with different capacities and explore different training schemes including both multi-task and single-task learning. Our results show that supervising the segmentation task improves our network's disparity estimation accuracy. We demonstrate a domain adaptation scheme where we supervise the segmentation task with monocular data and achieve domain adaptation of the adjacent disparity task, reducing disparity End-Point-Error and depth mean absolute error by 77.73% and 61.73% respectively compared to the pre-trained baseline model. Our best overall multi-task model, trained with both disparity and segmentation data in subsequent phases, achieves 89.15% mean Intersection-over-Union in RIS and 3.18 millimetre depth mean absolute error in SCARED test sets. Our proposed multi-task architecture is real-time, able to process (1280x1024) stereo input and simultaneously estimate disparity maps and segmentation masks at 22 frames per second. The model code and pre-trained models are made available: https://github.com/dimitrisPs/msdesis.