Given a scheme $Z$ over $X$, one can consider the sheaf of "sections $X\to Z$". More precisely, that sheaf associates to every étale open $U\to X$ the set of commutative diagrams
$$
\begin{matrix}
& & Z \\\
& \nearrow & \downarrow \\\
U & \to & X
\end{matrix}
$$
The scheme $\tilde Z$ you are looking for is the <i>espace &eacute;tal&eacute;</i> of the above sheaf, a <s>highly non-separated</s> scheme.