= Solution
The real part of a <holomorphic function> is real analytic and has a local complexification in its two real coordinates. Define $f^*(w)=\overline{f(\bar w)}$. On real $(x,y)$,
$$
2u(x,y)=f(x+iy)+f^*(x-iy),
$$
and the same identity holds after analytic continuation to complex $(x,y)$. Substituting $x=(z+\bar z_0)/2$ and $y=(z-\bar z_0)/(2i)$ gives $x+iy=z$ and $x-iy=\bar z_0$, hence
$$
\boxed{f(z)=2u\left(\frac{z+\bar z_0}{2},\frac{z-\bar z_0}{2i}\right)-\overline{f(z_0)}.}
$$
The conjugation on the final constant is present in the PDF but was lost in the converted TeX. Without it the formula already fails when $z=z_0$ and $f(z_0)$ is not real.
Now $ze^{iz}=(x+iy)e^{-y}(\cos x+i\sin x)$ has the specified real part. Thus the complete family is
$$
\boxed{f(z)=ze^{iz}+iC,\qquad C\in\mathbb R.}
$$
Any two such <holomorphic functions> differ by a <holomorphic function> with zero real part; the <Cauchy-Riemann equations> make its imaginary part constant on a connected neighborhood.
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