Example of non-isomorphic sheaves, with isomorphic stalks at every point? Basically what the title asks.  I'd like to see an example of two sheaves on a topological space which have the same (isomorphic) stalks at every point, but are not isomorphic as sheaves.  

Take as the topological space $S^1$ the circle. Take as stalks, $\mathbb{Z}_3$. There is an automorphism switching the two non identity elements. One sheaf is the trivial sheaf, the other is twisted like a Möbius strip. 
