  We consider macroscopically large 3-partitions $(A,B,C)$ of connected
subsystems $A\cup B \cup C$ in infinite quantum spin chains and study the
R\'enyi-$\alpha$ tripartite information $I_3^{(\alpha)}(A,B,C)$. At equilibrium
in clean 1D systems with local Hamiltonians it generally vanishes. A notable
exception is the ground state of conformal critical systems, in which
$I_3^{(\alpha)}(A,B,C)$ is known to be a universal function of the cross ratio
$x=|A||C|/[(|A|+|B|)(|C|+|B|)]$, where $|A|$ denotes $A$'s length. We identify
different classes of states that, under time evolution with translationally
invariant Hamiltonians, locally relax to states with a nonzero (R\'enyi)
tripartite information, which furthermore exhibits a universal dependency on
$x$. We report a numerical study of $I_3^{(\alpha)}$ in systems that are dual
to free fermions, propose a field-theory description, and work out their
asymptotic behaviour for $\alpha=2$ in general and for generic $\alpha$ in a
subclass of systems. This allows us to infer the value of $I_3^{(\alpha)}$ in
the scaling limit $x\rightarrow 1^-$, which we call ``residual tripartite
information''. If nonzero, our analysis points to a universal residual value
$-\log 2$ independently of the R\'enyi index $\alpha$, and hence applies also
to the genuine (von Neumann) tripartite information.
