Let $X$ and $Y$ be uncountable Polish spaces. We show in ZF that there is a coanalytic subset $P$ of $X \times Y$ with countable sections which cannot be expressed as the union of countably many partial coanalytic, or even $\mathrm{PCA} = \Sigma^1_2$, graphs. If $X = Y = \omega^\omega, P$ may be taken to be $\Pi^1_1$. Assuming stronger set theoretic axioms, we identify the least pointclass such that any such coanalytic $P$ can be expressed as the union of countably many graphs in this pointclass. This last result is extended (under suitable hypotheses) to all levels of the projective hierarchy.