  Topology is central to phenomena that arise in a variety of fields, ranging
from quantum field theory to quantum information science to condensed matter
physics. Recently, the study of topology has been extended to open systems,
leading to a plethora of intriguing effects such as topological lasing,
exceptional surfaces, as well as the non-Hermitian bulk-boundary
correspondence. Here, we show that resonator networks with dissipative
couplings can be governed by matrix-valued modified Wilson lines, leading to
non-Abelian effects. This is in contrast to conservative Hamiltonians
exhibiting non-degenerate energy levels, where the geometric properties of the
Bloch eigenstates are typically characterized by scalar Berry phases. We
experimentally measure geometric phases and demonstrate non-Abelian effects in
a dissipatively-coupled network of time-multiplexed photonic resonators. Our
results point to new ways in which the combined effect of topology and
engineered dissipation can lead to non-Abelian topological phenomena.
