For an $n$$2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator d$\overline{\partial}$, built locally from the usual exterioranti-holomorphic derivative, that acts on $\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{\bullet}(M)$$\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{(0,\bullet)}(M)$ so as to give a complex $$ \Gamma^{\infty}(E) \overset{d}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{1}(M) \overset{d}{\to} \cdots \overset{d}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{n}(M)\overset{d}{\to} 0. $$$$ \Gamma^{\infty}(E) \overset{\overline{\partial}}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{(0,1)}(M) \overset{\overline{\partial}}{\to} \cdots \overset{\overline{\partial}}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{(0,n)}(M)\overset{\overline{\partial}}{\to} 0. $$
As I prefer global constructions, I began to wonder how one would construct this globally. To apply id $\otimes$ d$\otimes \overline{\partial}$ to $\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{\bullet}(M)$$\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{(0,\bullet)}(M)$ is of course not well-defined since we are tensoring over ${C^{\infty}(M)}$. So what can one do . . .?
P.S. Does such a complex exist in the purely real case, and if not, then why not?