$p$-adic Kato--Nakayama space Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this construction (presumably something rigid-analytic)?
 A: I believe many people have thought about this at some point, and I don't think such a construction is known. 
In the paper https://arxiv.org/abs/1207.3380 , Yves Andre considers the real blowups (in the complex plane) as a completion with respect to the "sectorial uniformity", and then considers some $p$-adic or non-archimedean variants. I am not sure if this produces a desired analog of the Kato-Nakayama space. Andre's point of view is applications to singularities of differential equations (especially irregular, i.e. worse than logarithmic poles), phenomena like overconvergence etc.
There are the algebraic analogs of the Kato-Nakayama space: the (Kummer) etale site or the infinite root stack, both equipped with a "proper" projection down to the underlying scheme, which in the log regular case are "homotopy equivalent" to the trivial locus of the log structure (in characteristic zero, or after prime to $p$ completion), and which over the complex numbers can be compared to the Kato-Nakayama space (see e.g. the paper of Carchedi-Scherotzke-Sybilla-Talpo).
