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Macroscale chains have been proposed to give insight into the physics of molecular polymer systems. Nevertheless, understanding the rheological response of systems of quasi-one-dimensional semiflexible materials, such as bead-chain packings, is currently a great challenge. We study the nonlinear rheology of random assemblies of macroscale chains -- including steel bead chains and cooked spaghetti -- under oscillatory shear. We show that a universal transition from localized to wide shear zones occurs upon increasing the strain amplitude, for a wide range of lengths, flexibilities, and other structural parameters of the constituent elements. The critical strain amplitude coincides with the onset of strain stiffening development in the system. We obtain scaling laws for transition sharpness, shear-zone width, and stiffness enhancement as a function of chain length. Our findings suggest that the entanglements between the constituent elements strengthen when approaching the critical strain amplitude and rapidly become long range, even spanning the entire finite system for long enough chains. We show that the nonlinear rheological response is governed by the interplay between increasing stored elastic forces due to entanglements and increasing contribution of dissipation with shear rate and interlocking between chains.