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 Rényi-$α$ tripartite information $I_3^{(α)}(A,B,C)$. 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 $I_3^{(α)}(A,B,C)$ 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 (Rényi) tripartite information, which furthermore exhibits a universal dependency on $x$. We report a numerical study of $I_3^{(α)}$ in systems that are dual to free fermions, propose a field-theory description, and work out their asymptotic behaviour for $α=2$ in general and for generic $α$ in a subclass of systems. This allows us to infer the value of $I_3^{(α)}$ 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 Rényi index $α$, and hence applies also to the genuine (von Neumann) tripartite information.