In Lectures on Analytic Geometry, for complex-analytic geometry, seemingly one only considers maps $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ of analytic rings for $0<p\le1$ where $A$ is a "structure ring" like $\mathcal O(\overline D)$ and $M$ is induced by the underlying map of condensed rings. In other words, $\mathcal A$ could be seen as an algebra in $D_{\ge0}(\mathbb C,\mathcal M_{<p})$ (more precisely, a module over the symmetric monad) and $D_{\ge0}(\mathcal A,\mathcal M)$ is simply the $\infty$-category of modules over $\mathcal A$ as an algebra in $D_{\ge0}(\mathbb C,\mathcal M_{<p})$, i.e. no further analytic structure is imposed.

I wonder whether there are "natural" examples of $(\mathbb C,\mathcal M_{<p})\to(\mathcal A,\mathcal M)$ where $\mathcal M$ is not induced by the map $\mathbb C\to\mathcal A$, just as in the rigid analytic geometry, we have finer concepts like Huber pairs?

There is a related question: in Liquid tensor experiment Remark 5.5, we have $p$-liquid analytic rings $(\mathbb Q_\ell,\mathcal M_{<p})$ for all $p\in(0,+\infty]$. When $p=+\infty$, it is the solid theory. I wonder whether this $p$-liquid picture generalizes to Huber pairs? If this is possible, then it might be reasonable to imagine relating the two cases via $\mathbb Z[T]$.