Suppose $F:Sch^\text{op}\to Set$ is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must $F$ be an algebraic space? That is, must $F$ have an étale cover by a scheme?

Note that it is enough to find an fppf cover of $F$ by a scheme, at which point you know that $F$ is an algebraic stack ("Artin's slice theorem", 10.1 of Laumon-Moret-Bailly) fibered in sets, so it's an algebraic space.

The analogous question, "is every fpqc stack an algebraic stack?" has a negative answer, but the counterexample is "purely stacky".