  We give a treatment of relative Calabi--Yau structures on functors between
$R$-linear stable $\infty$-categories, with $R$ any $\mathbb{E}_\infty$-ring
spectrum, generalizing previous treatments in the setting of dg-categories.
Using their gluing properties, we further construct relative Calabi--Yau
structures on the global sections of perverse schobers, i.e. categorified
perverse sheaves, on surfaces with boundary. We treat examples related to
Fukaya categories and representation theory. In a related direction, we define
the monodromy of a perverse schober parametrized by a ribbon graph on a framed
surface and show that it forms a local system of stable $\infty$-categories.
