  We show that a $\mathbb{Z}_3$ quantum double can be realized in an array of
superconducting wires coupled via Josephson junctions. With a suitably chosen
magnetic flux threading the system, the inter-wire Josephson couplings take the
form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry
-- a local $\mathbb{Z}_3$ symmetry involving permutations and shifts by $\pm
2\pi/3$ of the superconducting phases. The sign of the star potential resulting
from the Josephson energy is inverted in this physical realization, leading to
a massive degeneracy in the non-zero flux sectors. A dimerization pattern
encoded in the capacitances of the array lifts up these degeneracies, resulting
in a $\mathbb{Z}_3$ topologically ordered state. Moreover, this dimerization
pattern leads to a larger effective vison gap as compared to the canonical case
with the usual (uninverted) star term. We further show that our model maps to a
quantum three-state Potts model under a duality transformation. We argue, using
a combination of bosonization and mean field theory, that altering the
dimerization pattern of the capacitances leads to a transition from the
$\mathbb{Z}_3$ topological phase into a quantum XY-ordered phase. Our work
highlights that combinatorial gauge symmetry can serve as a design principle to
build quantum double models using systems with realistic interactions.
