  We study the following question: Let $(X,g)$ be a compact Gauduchon surface,
$(E,h)$ be a differentiable rank $r$ vector bundle on $X$, ${\mathcal{D}}$ be a
fixed holomorphic structure on $D:=\det(E)$ and $a$ be the Chern connection of
the pair $(\mathcal{D},\det(h))$. Does the complex space structure on
${\mathcal{M}}_a^{\mathrm{ASD}}(E)^*$ induced by the Kobayashi-Hitchin
correspondence extend to a complex space structure on the Donaldson-Uhlenbeck
compactification $\overline{\mathcal{M}}_a^\mathrm{ASD}(E)$? Our results answer
this question in detail for the moduli spaces of $\mathrm{SU}(2)$-instantons
with $c_2=1$ on general (possibly unknown) class VII surfaces.
