  Non-Abelian topological phases (NATPs) are highly sought-after candidate
states for quantum computing and communication while lacking straightforward
configuration and manipulation, especially for classical waves. In this work,
we exploit novel braid-type couplings among a pair of triple-component acoustic
dipoles, which act as functional elements with effective imaginary couplings.
Sequencing them in one dimension allows us to generate acoustic NATPs in a
compact yet reciprocal Hermitian system. We further provide the whole phase
diagram that encompasses all i, j, and k non-Abelian phases, and directly
demonstrate their unique quotient relations via different endpoint states. Our
NATPs based on real-space braiding may inspire the exploration of acoustic
devices with non-commutative characters.
