I have heard that there exist examples of Huber pairs $(A,A^{\circ})$ and $(B,B^+)$ such that $\operatorname{Spa}(B,B^+)$ is a rational open of $\operatorname{Spa}(A,A^{\circ})$ and such that $B^+ \neq B^{\circ}$. I'm looking for an example of this, say for $A$ Tate and sheafy.
This phenomenon seems to be fundamental for the subject: it is one reason why the theory is developed with pairs of rings. But I wasn't able to find an explicit example in the references I know, only the assertion that they exist.
It is known that such an example cannot occur if $A$ is the ring of functions on an affinoid rigid analytic variety. Are there obvious larger classes of rings for which this phenomenon cannot occur? Is the result true if $\operatorname{Spa}(A)$ is pro-étale over an affinoid rigid analytic variety? Can such examples occur if $A$ is perfect of characteristic $p$?
