Automatic calibration of multi-camera systems, namely the accurate estimation of spatial extrinsic parameters, is fundamental for 3D reconstruction, panoramic perception, and multi-view data fusion. Existing methods typically rely on calibration targets, explicit geometric modeling, or task-specific neural networks. Such approaches often exhibit limited robustness and applicability in complex dynamic environments or online scenarios, making them difficult to deploy in practical applications. To address this, this paper proposes GMAC, a multi-camera extrinsic estimation framework based on the implicit geometric representations learned by multi-view reconstruction networks. GMAC models extrinsics as global variables constrained by the latent multi-view geometric structure and prunes and structurally reconfigures existing networks so that their latent features can directly support extrinsic prediction through a lightweight regression head, without requiring a completely new network design. Furthermore, GMAC jointly optimizes cross-view reprojection consistency and multi-view cycle consistency, ensuring geometric coherence across cameras while improving prediction accuracy and optimization stability. Experiments on both synthetic and real-world multi-camera datasets demonstrate that GMAC achieves accurate and stable extrinsic estimation without explicit 3D reconstruction or manual calibration, providing a new solution for efficient deployment and online calibration of multi-camera systems.