This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a finitely-generated saturated (a.k.a. normal) monoid $\overline{M}_x=M_x/\mathcal{O}_x^\times,$ which is in some sense "responsible" for the log structure at $x$.
I want to give a name to the subcategory of log varieties for which $\overline{M}_x$ is free, i.e., isomorhpic to $\mathbb{N}^k$ at every point (here $k$ can depend on the point). If $(X,M)$ is a smooth log variety, this condition is equivalent (at least in characteristic $0$) to the defining divisor of $M$ being normal crossings. In the past, I've called varieties "of normal crossings type", but I'm not so happy with the name, since it suggests smoothness whereas many non-smooth log varieties (such as the log variety on a point with $\bar{M} = \mathbb{N}^k$) have this property.
Other names I've considered are "regular" (but "regular monoid" means something else) and "cartesian" (but that conflicts with terms in algebraic geomtery and category theory). Is there a standard name for such varieties?
