  We study the resurgent structure of the refined topological string partition
function on a non-compact Calabi-Yau threefold, at large orders in the string
coupling constant $g_s$ and fixed refinement parameter $\mathsf{b}$. For
$\mathsf{b}\neq 1$, the Borel transform admits two families of simple poles,
corresponding to integral periods rescaled by $\mathsf{b}$ and $1/\mathsf{b}$.
We show that the corresponding Stokes automorphism is expressed in terms of a
generalization of the non-compact quantum dilogarithm, and we conjecture that
the Stokes constants are determined by the refined Donaldson-Thomas invariants
counting spin-$j$ BPS states. This jump in the refined topological string
partition function is a special case (unit five-brane charge) of a more general
transformation property of wave functions on quantum twisted tori introduced in
earlier work by two of the authors. We show that this property follows from the
transformation of a suitable refined dual partition function across BPS rays,
defined by extending the Moyal star product to the realm of contact geometry.
