Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over the category of complex manifolds? 


Obviously, this is possible for a stack represented by the quotient of a complex manifold by a group action, so I'd imagine that this should at least be possible for DM stacks, but I can't think of any general construction that doesn't involve some kind of atlas.



how to construct a $C^\infty$ stack from a holomorphic stack
Denote by $$u:CxMfd \to Mfd$$ the forgetful functor from complex manifolds to smooth manifolds. Let $$u\_!:St\left(CxMfd\right) \to St\left(Mfd\right)$$ denote its 2-categorical prolongation. Explicitly, this is given by the bicategorical Kan extension of $y\_{Mfd} \circ u$ along the Yoneda embedding $$y\_{CxMfd}:CxMfd \to St\left(CxMfd\right),$$ where $y\_{Mfd}$ is similarly defined. $u\_!$ is the unique weak colimit preserving functor which agrees with $y\_{Mfd} \circ u$ on representables.


I claim that $u\_!$ sends holomorphic stacks (stacks coming from groupoid objects in complex manifolds) to differentiable stacks.


Indeed, let $\mathcal{X}$ be a holomorphic stack coming from a groupoid object $X\_1 \rightrightarrows X\_0.$ Then, $\mathcal{X}$ is the weak colimit of the truncated semi-simplicial diagram $$X\_2\mspace{5mu} \{(3\mspace{5mu} parallel \mspace{5mu} arrows)\}\mspace{5mu} X\_1 \rightrightarrows X\_0,$$ viewing each $X\_i$ as a representable presheaf on $CxMfd$. Applying $u\_!$ to this diagram, yields that $u\_!\left(\mathcal{X}\right)$ is the weak colimit of the same diagram, now viewing each $X\_i$ as a representable presheaf in $Mfd$. This in turn implies that $u\_!\left(\mathcal{X}\right)$ is the stackification of the weak presheaf of groupoids arising canonically from $X$ viewed as a Lie groupoid. In particular, this implies that $u\_!$, when restricted to holomorphic stacks agrees with the answer of David Roberts, only, it makes no explicit reference to atlases.


