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.

`pullback' objects and morphisms to your original stack from the extra objects and morphisms in the category of manifolds, then somehow take the stack`

generated' by what you get. – Brett Parker Apr 12 '12 at 23:59