AbstractBloch points are small 3D magnetic textures with unit topological charge that require the use of advanced vector imaging techniques for their direct experimental characterization. Here we show results from the reconstruction of the magnetization field $${{{{{\bf{m}}}}}}\left({{{{{\bf{r}}}}}}\right)$$ m r of an elongated permalloy microstructure by X-ray vector magnetic tomography. A central asymmetric Bloch domain wall is observed, decorated by Bloch points arranged in several dipoles and a triplet. The analysis of the $${{{{{\bf{m}}}}}}\left({{{{{\bf{r}}}}}}\right)$$ m r map in terms of topological concepts provides a quantitative description of the Bloch points as topological monopoles connected by bundles of emergent magnetic field lines carrying a fractional topological flux. It also reveals the topological constraints that determine chirality transitions of the central domain wall in the vicinity of Bloch points, independent of specific material properties. This approach could be readily extended to the study of magnetic microstructures in arbitrary remanent configurations.