  Given a bounded regular domain $\omega \subset \mathbb{R}^{N-1}$ and the
half-cylinder $\Sigma = \omega \times (0,+\infty)$, we consider the relative
overdetermined torsion problem in $\Sigma$, i.e. \[\left\{ \begin{array}{ll}
\Delta {u}+1=0 &\mbox{in $\Omega$},\newline \partial_\eta u = 0 &\mbox{on
$\widetilde \Gamma_\Omega$},\newline u=0 &\mbox{on $\Gamma_\Omega$},\newline
\partial_{\nu}u =c &\mbox{on $\Gamma_\Omega$}. \end{array} \right.
  \] where $\Omega \subset \Sigma$, $\Gamma_\Omega = \partial \Omega \cap
\Sigma$, $\widetilde \Gamma_\Omega = \partial \Omega \setminus \Gamma_\Omega$,
$\nu$ is the outer unit normal vector on $\Gamma_\Omega$ and $\eta$ is the
outer unit normal vector on $\widetilde \Gamma_\Omega$. We build nontrivial
solutions to this problem in domains $\Omega$ that are the hypograph of certain
nonconstant functions $v : \overline{\omega} \to (0, + \infty)$. Such solutions
can be reflected with respect to $\omega$, giving nontrivial solutions to the
relative overdetermined torsion problem in a cylinder. The proof uses a local
bifurcation argument which, quite remarkably, works for any generic base
$\omega$.
