It is well-known (Hartshorne 2.1.1) that if $F$ and $G$ are sheaves on a space $X$, then $\phi:F\rightarrow G$ is an isomorphism if and only if the induced stalk map $\phi_p:F_p\rightarrow G_p$ is an isomorphism for all $p\in X$. However, if we have a collection of isomorphisms $\{\psi_p:F_p\rightarrow G_p\}_{p\in X}$, this does not guarantee that $F$ and $G$ are isomorphic, because the $\psi_p$ might not be related to each other, i.e. there might not be a sheaf map $\psi:F\rightarrow G$ such that $\psi_p$ is the induced stalk map for all $p\in X$.

However, I was recently making this point to someone and was unable to think of a good example of non-isomorphic $F$ and $G$ having isomorphisms $\psi_p:F_p\rightarrow G_p$. I'm sure I knew one at some point, but I'm blanking on it now. Can someone provide an illustrative example, e.g. an example that occurs in some natural or basic problem, or one that captures the essential pattern of any example where this issue arises, or one where it is clear that $F$ and $G$ could not be isomorphic?