The Brillouin Klein bottle induced by ${\mathbb{Z}}_2$ gauge field has been recently proposed and has sparked growing interest. The momentum-space nonsymmorphic symmetry that it respects fundamentally modifies the Bloch band theory and expands the topological classification. Previous works limited to systems under ${\mathbb{Z}}_2$ gauge field have been developed, leaving the interaction between non-Abelian gauge fields and symmetries untouched. Here, we report theoretical and experimental realization of entwined Klein bottles with two orthogonal sub-bottles induced by $U_2$ non-Abelian gauge field, which stems from the hopping between two inseparable pseudospins. The bilayer acoustic crystals with non-Abelian coupling are constructed here to explore different topologies from the traditional Klein bottles under the Abelian fields. This research sheds light on both first-order and higher-order topologies unexplored in non-Abelian gauge fields, and enriches the topological classification with gauge symmetry beyond the existing paradigms.

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• Acoustics
• Non-Abelian gauge theories
• Topological effects in acoustic systems
• Acoustic metamaterials