Note that constant $F$ works. Otherwise, assuming $F > 0,$ your condition is just $ \Delta \sqrt F = 0. $ So you want some positive harmonic function $G$ and then $F = G^2.$ Similar for $F < 0, \; F = - H^2, \; \; \Delta \sqrt {-F} = 0.$
From Liouville's theorem, if you want a solution on all of $\mathbb R^3$ then $G$ and $F$ are constant. So, if you start with a nonconstant harmonic $G,$ such as linear, along a finite set of surfaces $G$ actually becomes $0$ and everything goes sideways.
EDIT, Thursday: Looking again, it is not so bad when the function is $0,$ as long as we are squaring, so the gradient is also $0.$ So, a solution, and possibly all solutions, are some real constant $c$ and harmonic $W,$ then $ F = c W^2. $