  We reinterpret a result of T. Mochizuki about the Fourier transform of Stokes
data of irregular connections on the Riemann sphere in the language of Stokes
local systems due to P. Boalch. We thus obtain a clean topological description
of the Stokes matrices of the Fourier transform from infinity to infinity in a
large number of cases of one level. In particular, this induces explicit
isomorphisms between wild character varieties, in a much larger range of
examples than those for which such isomorphisms have previously been written
down. We conjecture that these isomorphisms are compatible with the
quasi-Hamiltonian structure of the wild character varieties.
