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Quasicrystals possess collective phason degrees of freedom whose natural configuration space is the tiling hull and its singular strata rather than ordinary physical space. This paper constructs a topological framework to determine when closed phason cycles induce non-Abelian unitaries on the topologically degenerate Hilbert spaces of two-dimensional intrinsic topological orders. The central object of study is a projective adiabatic representation $\rho_{\text{ph}}: \pi_1(\Omega_{\text{QC}} \setminus \Delta, \lambda_0) \to \text{PU}(\mathcal{H}_{\text{top}})$, defined over a finite-local-complexity tiling hull or approximant configuration space $\Omega_{\text{QC}}$ with a discriminant locus $\Delta$ removed. By evaluating a hierarchy of models spanning toric code, quantum double $D(S_3)$, and Fibonacci Levin-Wen phases, we establish that defect-free local phason motion related by natural string-net recellulations yields at most an Abelian Berry phase. Conversely, non-Abelian holonomy arises when phason cycles enclose defects carrying nontrivial braided autoequivalences of the underlying anyon category, acting on Wilson-line algebras as twist-defect monodromies.