  `With persistence, a drop of water hollows out the stone' goes the ancient
Greek proverb. Yet, canonical percolation models do not account for
interactions between a moving tracer and its environment. Recently, we have
introduced the Sokoban model, which differs from this convention by allowing a
tracer to push single obstacles that block its path. To test how this newfound
ability affects percolation, we hereby consider a Bethe lattice on which
obstacles are scattered randomly and ask for the probability that the Sokoban
percolates through this lattice, i.e., escapes to infinity. We present an exact
solution to this problem and determine the escape probability as a function of
obstacle density. Similar to regular percolation, we show that the escape
probability undergoes a second-order phase transition. We exactly determine the
critical obstacle density at which this transition occurs and show that it is
higher than that of a tracer without obstacle-pushing abilities. Our findings
assert that pushing facilitates percolation on the Bethe lattice, as
intuitively expected. This result, however, sharply contrasts with our previous
findings on the 2D square lattice. There, the Sokoban cannot escape
$\unicode{x2013}$ not even at densities well below the percolation threshold.
We discuss the reasons behind this striking difference, which calls for a
deeper and better understanding of percolation in the presence of tracer-media
interactions.
