Efficient and accurate feed-forward multi-view reconstruction has long been an important task in computer vision. Recent transformer-based models like VGGT and $\pi^3$ have achieved impressive results with simple architectures, yet they face an inherent runtime bottleneck, due to the quadratic complexity of the global attention layers, that limits the scalability to large image sets. In this paper, we empirically analyze the global attention matrix of these models and observe that probability mass concentrates on a small subset of patch-patch interactions that correspond to cross-view geometric matches. Motivated by the structured attention and inspired by recent advancement in large language models, we propose a replacement for the dense global attention operation based on highly optimized block-sparse kernels, yielding up to $4\times$ faster inference with comparable task performance. Our retrofit requires no retraining of the backbone, extends to both VGGT and $\pi^3$, and supports large image collections. Evaluations on a comprehensive suite of multi-view benchmarks demonstrate the effectiveness of our approach.