  We analyse the chiral symmetry in the random $\pm J$ $XY$ model on a $N\times
2$ square lattice with periodic boundary conditions in the transverse
direction. This ``tube" lattice may be seen as a two-dimensional lattice of
which one dimension has been compactified. In the Villain formulation the
discrete-valued {\em chiralities}\/ or {\em charges}\/ associated with the
plaquettes of the lattice decouple from the continuous degrees of freedom. The
difficulty of the problem lies in the fact that the chiralities interact
through the long range ``strong" one-dimensional Coulomb potential - which
increases linearly with distance - as well as through an exponentially decaying
``weak" interaction. By comparing the ground state energies for periodic,
antiperiodic, and reflecting boundary conditions in the longitudinal direction,
we show that the chiralities and the $XY$ spins have the {\em same}\/ zero-$T$
correlation length exponent, whose exact value $\nu_c = 0.5564\ldots$ we
determine. The equality of these correlation lengths even in the presence of
long range chirality-chirality interactions lends support to the view that
chiral glass order cannot be sustained without simultaneous spin glass order.
