  Let $\Omega$ be a complex manifold, and let $X\subset \Omega$ be an open
submanifold whose closure $\bar X$ is a (not necessarily compact) submanifold
with smooth boundary. Let $G$ be a complex Lie group, $\Pi$ be a differentiable
principal $G$-bundle on $\Omega$ and $J$ a formally integrable bundle almost
complex structure on the restriction $\bar P:= \Pi|_{\bar X}$. We prove that,
if the boundary of $\bar X$ is strictly pseudoconvex, $J$ extends to a
holomorphic structure on the restriction of $\Pi$ to a neighborhood of $\bar X$
in $\Omega$. This answers positively and generalizes a problem stated in the
article "Boundary value problems for Yang-Mills fields" by S. Donaldson. We
obtain a gauge theoretical interpretation of the quotient
$\mathcal{C}^\infty(\partial \bar X,G)/\mathcal{O}^\infty(\bar X,G)$ associated
with any compact Stein manifold with boundary $\bar X$ endowed with a Hermitian
metric. For a fixed differentiable $G$-bundle $\bar P$ on a complex manifold
$\bar X$ with non-pseudoconvex boundary, we study the set of formally
integrable almost complex structures on $\bar P$ which admit formally
holomorphic local trivializations at boundary points. We give an example where
a "generic" formally integrable almost complex on $\bar P$ admits formally
holomorphic local trivializations at no boundary point, whereas the set of
formally integrable almost complex structures which admit formally holomorphic
local trivializations at all boundary points is dense.
