= Solution
Take the <punctured complex plane>
$$
D=\mathbb C^*=\mathbb C\setminus\{0\}
$$
and
$$
u(z)=\log|z|.
$$
In <polar coordinates>, $u(r,\theta)=\log r$, so
$$
\Delta u=u_{rr}+\frac1r u_r+\frac1{r^2}u_{\theta\theta}
=-\frac1{r^2}+\frac1{r^2}=0.
$$
Thus $u$ is harmonic on $D$.
If $u=\operatorname{Re}f$ for a holomorphic function $f$, the calculation from part (a) would give
$$
f'(z)=u_x-iu_y=\frac1z.
$$
The integral of a derivative around a closed curve is zero, whereas the <residue theorem> gives
$$
\int_{|z|=1}\frac{dz}{z}=2\pi i.
$$
This contradiction proves that no such $f$ exists. It is precisely the <log modulus has no global harmonic conjugate on the punctured plane> obstruction.
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