  Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module over a complex
manifold $X$, and let $G$ be a vector bundle on $X$. We describe an explicit
isomorphism between two different representations of the global
$\DeclareMathOperator{\Ext}{Ext}\Ext$ groups
$\DeclareMathOperator{\Ext}{Ext}\Ext^k(\mathcal{F},G)$. The first
representation is given by the cohomology of a twisted complex in the sense of
Toledo and Tong, and the second one is obtained from the Dolbeault complex
associated with $G$. A key tool that we introduce for explicitly describing
this isomorphism is a residue current associated with a twisted resolution of
$\mathcal{F}$.
