  We simulate the two-dimensional XY model in the flow representation by a
worm-type algorithm, up to linear system size $L=4096$, and study the geometric
properties of the flow configurations. As the coupling strength $K$ increases,
we observe that the system undergoes a percolation transition $K_{\rm perc}$
from a disordered phase consisting of small clusters into an ordered phase
containing a giant percolating cluster. Namely, in the low-temperature phase,
there exhibits a long-ranged order regarding the flow connectivity, in contrast
to the qusi-long-range order associated with spin properties. Near $K_{\rm
perc}$, the scaling behavior of geometric observables is well described by the
standard finite-size scaling ansatz for a second-order phase transition. The
estimated percolation threshold $K_{\rm perc}=1.105 \, 3(4)$ is close to but
obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition
point $K_{\rm BKT} = 1.119 \, 3(10)$, which is determined from the magnetic
susceptibility and the superfluid density. Various interesting questions arise
from these unconventional observations, and their solutions would shed lights
on a variety of classical and quantum systems of BKT phase transitions.
