  A supertube is a supersymmetric configuration in string theory which occurs
when a pair of branes spontaneously polarizes and generates a new dipole charge
extended along a closed curve. The dipole charge of a codimension-2 supertube
is characterized by the U-duality monodromy as one goes around the supertube.
For multiple codimension-2 supertubes, their monodromies do not commute in
general. In this paper, we construct a supersymmetric solution of
five-dimensional supergravity that describes two supertubes with such
non-Abelian monodromies, in a certain perturbative expansion. In supergravity,
the monodromies are realized as the multi-valuedness of the scalar fields,
while in higher dimensions they correspond to non-geometric duality twists of
the internal space. The supertubes in our solution carry NS5 and 5$_2^2$ dipole
charges and exhibit the same monodromy structure as the SU(2) Seiberg-Witten
geometry. The perturbative solution has $\mathrm{AdS}_2\times S^2$ asymptotics
and vanishing four-dimensional angular momentum. We argue that this solution
represents a microstate of four-dimensional black holes with a finite horizon
and that it provides a clue for the gravity realization of a pure-Higgs branch
state in the dual quiver quantum mechanics.
