A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.

See for instance Defn 2.6 here.

What is the (geometric) significance of requiring $R^+ $ to be integrally closed?

On a related note, what is the difference (geometrically) between total integral closure and integrally closed. It is surprisingly hard to find anything about total integral closures...