Purpose: Phase singularities are a common problem in image reconstruction with auto-calibrated sensitivities due to an inherent ambiguity of the estimation problem. The purpose of this work is to develop a method for detecting and correcting phase poles in non-linear inverse (NLINV) reconstruction of MR images and coil sensitivity maps. Methods: Phase poles are detected in individual coil sensitivity maps by computing the curl in each pixel. A weighted average of the curl in each coil is computed to detect phase poles. Phase pole detection and correction is then integrated into the iteratively regularized Gauss-Newton method of the NLINV algorithm, which then avoid phase singularities in the reconstructed images. The method is evaluated for reconstruction of accelerated Cartesian MPRAGE data of the brain and interactive radial real-time MRI of the human heart. Results: Phase poles are reliably removed in NLINV reconstructions for both applications. NLINV with phase pole correction can reliably and efficiently estimate coil sensitivity profiles free from singularities even from very small ($7\times7$) auto-calibration (AC) regions. Conclusion: NLINV emerges as an efficient and reliable tool for image reconstruction and coil sensitivity estimation in challenging MRI applications.