We propose a novel fingerprint descriptor, namely Möbius moduli, measuring local deviation of orientation fields (OF) of fingerprints from conformal fields, and we propose a method to robustly measure them, based on tetraquadrilaterals to approximate a conformal modulus locally with one due to a Möbius transformation. Conformal fields arise by the approximation of fingerprint OFs given by zero-pole models, which are determined by the singular points and a rotation. This approximation is very coarse, e.g., for fingerprints with no singular points (arch type), the zero-pole model’s OF has parallel lines. Quadratic differential (QD) models, which are obtained from zero-pole models by adding suitable singularities outside the observation window, approximate real fingerprints much better. For example, for arch type fingerprints, parallel lines along the distal joint change slowly into circular lines around the nail furrow. Still, QD models are not fully realistic because, for example along the central axis of arch type fingerprints, ridge line curvatures usually first increase and then decrease again. It is impossible to model this with QDs, which, due to complex analyticity, also produce conformal fields only. In fact, as one of many applications of the new descriptor, we show, using histograms of curvatures and conformality indices (log of the absolute values of the Möbius moduli), that local deviation from conformality in fingerprints occurs systematically at high curvature which is not reflected by state-of-the-art fingerprint models as are used, for instance, in the well-known synthetic fingerprint generation tool SFinGe and these differences robustly discriminate real prints from SFinGe’s synthetic prints.