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Image registration is a fundamental medical image analysis task. Ideally, registration should focus on aligning semantically corresponding voxels, i.e., the same anatomical locations. However, existing methods often optimize similarity measures computed directly on intensities or on hand-crafted features, which lack anatomical semantic information. These similarity measures may lead to sub-optimal solutions where large deformations, complex anatomical differences, or cross-modality imagery exist. In this work, we introduce a fast and accurate method for unsupervised 3D medical image registration building on top of a Self-supervised Anatomical eMbedding (SAM) algorithm, which is capable of computing dense anatomical correspondences between two images at the voxel level. We name our approach SAM-Enhanced registration (SAME++), which decomposes image registration into four steps: affine transformation, coarse deformation, deep non-parametric transformation, and instance optimization. Using SAM embeddings, we enhance these steps by finding more coherent correspondence and providing features with better semantic guidance. We extensively evaluated SAME++ using more than 50 labeled organs on three challenging inter-subject registration tasks of different body parts. As a complete registration framework, SAME++ markedly outperforms leading methods by $4.2\%$ - $8.2\%$ in terms of Dice score while being orders of magnitude faster than numerical optimization-based methods. Code is available at \url{https://github.com/alibaba-damo-academy/same}.