As an exercise in understanding the notion of surjectivity in the category of sheaves, I came up with this example, slightly modifying the standard ones given in my textbooks. I feel like this one is easier to visualize, since it takes place in real spaces (as opposed to the standard examples, drawn from complex analysis, which I'm less familiar with).
I would appreciate verification and any thoughts on the construction.
The idea is to take the circle $\mathbb{S}^1$ with its standard smooth manifold structure. Then, to every point $x$, we designate a distinguished open $U_x$ defined as $\mathbb{S}^1 - \{\bar{x}\}$, where $\bar{x}$ is the antipodal point of $x$, and furthermore, we assign a distinguished chart $\pi :U_x \rightarrow \mathbb{R}$ by stereographic projection from $\bar{x}$. It can be checked that $\pi$ sends $x$ to $0$.
Then, we assign to each $U_x$ the ring of real analytic functions defined on $\pi(U_x)$. (Perhaps it's also important to stipulate that they have infinite radius of convergence?) Informally, each regular function on the circle is given by its Taylor series in the "$\theta$" coordinate.
With this setup of real analytic functions on the circle on a base, we ought to be able to extend this to a sheaf on the circle. (And without needing to sheafify).
The only global sections, then, should be those analytic functions which are $2\pi$-periodic. That is, when we transport a global section $f$ to $\pi(U_x)=\mathbb{R}$ and analytically continue it on the coordinate chart, we should find that it has a period of $2\pi$. Otherwise, as we "wrap it around" the circle, it will fail to match up, and the compatibility requirement in the gluing axiom will not be satisfied.
We can define a sheaf morphism $\phi$ by stipulating that $f(x) \in \mathscr{F}_x$ gets sent to $f(2x)$. We see that it sends each ($2\pi$-periodic) global section to a $\pi$-periodic global section. And so while the germs at every point are surjected on ($\phi(\sum c_n \frac{1}{2^n} \theta^n) = \sum c_n \theta^n$), there is no global section which maps to a global section with fundamental period $2\pi$.