Machine perception applications are increasingly moving toward manipulating and processing 3D point cloud. This paper focuses on point cloud registration, a key primitive of 3D data processing widely used in high-level tasks such as odometry, simultaneous localization and mapping, and 3D reconstruction. As these applications are routinely deployed in energy-constrained environments, real-time and energy-efficient point cloud registration is critical. We present Tigris, an algorithm-architecture co-designed system specialized for point cloud registration. Through an extensive exploration of the registration pipeline design space, we find that, while different design points make vastly different trade-offs between accuracy and performance, KD-tree search is a common performance bottleneck, and thus is an ideal candidate for architectural specialization. While KD-tree search is inherently sequential, we propose an acceleration-amenable data structure and search algorithm that exposes different forms of parallelism of KD-tree search in the context of point cloud registration. The co-designed accelerator systematically exploits the parallelism while incorporating a set of architectural techniques that further improve the accelerator efficiency. Overall, Tigris achieves 77.2$\times$ speedup and 7.4$\times$ power reduction in KD-tree search over an RTX 2080 Ti GPU, which translates to a 41.7% registration performance improvements and 3.0$\times$ power reduction.