  When traditional linearised theory is used to study gravity-capillary waves
produced by flow past an obstruction, the geometry of the object is assumed to
be small in one or several of its dimensions. In order to preserve the
nonlinear nature of the obstruction, asymptotic expansions in the low-Froude or
low-Bond number limits can be derived, but here, the solutions are waveless to
every order. This is because the waves are in fact, exponentially small, and
thus beyond-all-orders of regular asymptotics; their formation is a consequence
of the divergence of the asymptotic series and the associated Stokes
Phenomenon.
  In Part 1, we showed how exponential asymptotics could be used to study the
problem when the size of the obstruction is first linearised. In this paper, we
extend the analysis to the nonlinear problem, thus allowing the full geometry
to be considered at leading order. When applied to the classic problem of flow
over a step, our analysis reveals the existence of six classes of
gravity-capillary waves, from which two share a connection with the usual
linearised solutions first discovered by Lord Rayleigh. The new solutions arise
due to the availability of multiple singularities in the geometry, coupled with
the interplay of gravitational and cohesive effects.
