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Smoothly varying magnetization textures such as domain walls, skyrmions or hopfions serve as promising candidates for the information bits of the future. Understanding their physical properties is both a major field of interest and a theoretical challenge, involving the physics on different length scales. Here, we apply the phase space formulation of quantum mechanics to magnetic insulators and metals in the limit of zero temperature to obtain a gradient expansion in terms of real-space derivatives of the magnetization. Our primary focus is the anomalous Hall effect in noncollinear magnets which serves as an important proxy in the detection of localized magnetic structures. We formulate the problem in the language of noncommutative fiber bundles and make the central finding that the semiclassical expansion of the density matrix and the Berry curvature is governed by a construction from string theory which is known as the Seiberg-Witten map. Originally discovered in the effective low-energy behavior of D-branes, this map now gives a geometrical underpinning to gradient expansion techniques in noncollinear magnets and offers a radically new perspective on their electronic properties.