Dirichlet Problem: Example where the Green function is not the Poisson kernel

Give an example of a Dirichlet problem where the Green function is not the Poisson kernel.
For a bounded open domain $D$ with a sufficiently smooth boundary and $f \in C\left(\partial D \right)$, the Dirichlet problem is $$\Delta u=0 \text{, on } D ,\\ \left. u\right|_{\partial D}=f \text{, on } \partial D.$$

Following this answer, there should be such example.
In wikipedia they take the quarter plane $\{(x,y):x,y \ge 0\}$. But I can't redo this example.

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