  We interpret the support $\tau$-tilting complex of any gentle bound quiver as
the non-kissing complex of walks on its blossoming quiver. Particularly
relevant examples were previously studied for quivers defined by a subset of
the grid or by a dissection of a polygon. We then focus on the case when the
non-kissing complex is finite. We show that the graph of increasing flips on
its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we
study its $\mathbf{g}$-vector fan and prove that it is the normal fan of a
non-kissing associahedron.
