Loss of global phase coherence is the mechanism behind the Kosterlitz-Thouless (KT) transition in two dimensional systems. Yet the effect of disorder on the KT transition is unresolved. Here we model disordered superconducting thin films using the disordered classical two-dimensional XY model. We propose and simulate an experiment to probe the spatial structure of the KT transition in such disordered systems by studying the global effect of cutting individual bonds. We present evidence that near the transition, global phase coherence propagates in specific channels and discuss the relation between such channels and a percolation of a near neighbor correlations.