  We show that, for pertinent values of the fabrication and control parameters,
an attractive finite coupling fixed point emerges in the phase diagram of a
Y-junction of superconducting Josephson chains. The new fixed point arises only
when the dimensionless flux $f$ piercing the central loop of the network equals
$\pi$ and, thus, does not break time-reversal invariance; for $f \neq \pi$,
only the strongly coupled fixed point survives as a stable attractive fixed
point. Phase slips (instantons) have a crucial role in establishing this
transition: we show indeed that, at
  $f = \pi$, a new set of instantons -the W-instantons- comes into play to
destabilize the strongly coupled fixed point.
  Finally, we provide a detailed account of the Josephson current-phase
relationship along the arms of the network, near each one of the allowed fixed
points. Our results evidence remarkable similarities between the phase diagram
accessible to a Y-junction of superconducting Josephson chains and the one
found in the analysis of quantum Brownian motion on frustrated planar lattices.
