Multi-view 3D reconstruction for reflective and textureless surfaces remains a challenging problem. Both camera pose calibration and shape reconstruction fail due to insufficient or unreliable visual features across views. To address these issues, we present PMNI (Pose-free Multiview Normal Integration), a novel neural surface reconstruction method that leverages surface normal maps instead of RGB images to incorporate rich geometric information. By enforcing geometric constraints from surface normals and multiview shape consistency within a neural signed distance function (SDF) optimization framework, PMNI robustly recovers both camera poses and high-fidelity surface geometry simultaneously. Experimental results on synthetic and real-world datasets show that our method achieves state-of-the-art performance in the reconstruction of reflective surfaces, even without reliable initial camera poses.