  We consider the R\'enyi-$\alpha$ tripartite information $I_3^{(\alpha)}$ of
three adjacent subsystems in the stationary state emerging after global
quenches in noninteracting spin chains from both homogeneous and bipartite
states. We identify settings in which $I_3^{(\alpha)}$ remains nonzero also in
the limit of infinite lengths and develop an effective quantum field theory
description of free fermionic fields on a ladder. We map the calculation into a
Riemann-Hilbert problem with a piecewise constant matrix for a doubly connected
domain. We find an explicit solution for $\alpha=2$ and an implicit one for
$\alpha>2$. In the latter case, we develop a rapidly convergent perturbation
theory that we use to derive analytic formulae approximating $I_3^{(\alpha)}$
with outstanding accuracy.
