  Scale-separated AdS compactifications of string theory can be constructed at
the two-derivative supergravity level in the presence of smeared orientifold
planes. The unsmearing corrections are known to leading order in the large
volume, weak coupling limit. However, first-order perturbative approximations
of non-linear problems can often produce spurious solutions, which are only
weeded out by additional consistency conditions imposed at higher orders. In
this work, we revisit the unsmearing procedure and present consistency
conditions obtained from the second order warp factor and dilaton equations.
This requires proper treatment of the near-source singularities. The resulting
conditions appear as integral constraints on various non-linear combinations of
the first order corrections, which we argue can generally be satisfied by
appropriate choice of integration constants of the leading-order solutions.
This provides a non-trivial consistency check for the perturbative unsmearing
procedure and supports the existence of scale-separated AdS vacua in string
theory.
