Living organisms form a large variety of hierarchically structured extracellular functional tissues. Remarkably, these materials exhibit regularity and structural coherence across multiple length scales, far beyond the size of a single cell. Here, synchrotron-based nanotomographic imaging in combination with machine-learning-based segmentation is used to reveal the structural synchronization process of nacre forming in the shell of the mollusc Unio pictorum. We show that the emergence of this highly regular layered structure is driven by a disorder-to-order transition achieved through the motion and interaction of screw-like structural dislocations with an opposite topological sign. Using an analogy to similar processes observed in liquid-crystalline systems, we demonstrate that these microstructural faults act as dissipative topological defects coupled by an elastic distortion field surrounding their cores. Their mutual annihilation results in structural synchronization that is simulated using the classical Kuramoto model. The developed experimental, theoretical and numerical framework provides a comprehensive physical view of the formation of biogenic materials.