Recently on glancing through Hartshorne's description of Cartier divisors I started pondering the definition of sheafification which led me to a question I can't answer. Neither can I find a discussion in the standard texts.

First let's set up some notation.
Let $\mathcal{F}$ be a presheaf (let's say of sets)
on the topological space $X$.
It has a sheafification $\mathcal{F}^+$. We want to describe
the global sections of $\mathcal{F}^+$.
A global section of $\mathcal{F}^+$ consists
of a map $f:x\mapsto f_x$ where $f_x\in\mathcal{F}_x$,
the stalk of $\mathcal{F}$ at the point $x\in X$. This map
$f$ also obeys a local compatibility condition: there is an open
covering $(U_i)_{i\in I}$ of $X$, and sections $g_i\in\mathcal{F}(U_i)$
with the property that whenever $x\in U_i$ then $f_x$
equals the germ of $g_i$ at the point $x$. Let's call such a covering
$(U_i)$ and sections $(g_i)$ a *representing system of sections* for
the global section. (Is there a standard term for this concept?)

My question is this:

For each global section of $\mathcal{F}^+$ is there always a representing system of sections for it having the stronger compatibility property that $$g_i|_{U_i\cap U_j}=g_j|_{U_i\cap U_j}\in\mathcal{F}(U_i\cap U_j)$$ for all $i$, $j\in I$? If not, is there some reasonable condition on the presheaf $\mathcal{F}$ that will guarantee this?

Another motivation is to find a good "pointless" description of the sheafification functor (in the sense of "pointless topology" or locale theory). The definitions of presheaf and sheaf only use the complete lattice structure on the collection of open sets of $X$ and so are thoroughly "pointless", but the usual description of the sheafification functor uses the definitely "pointy" notion of stalk.