Simplicial $*$-modules and mild actions

We develop an analogue of the theory of $*$-modules in the world ofsimplicial sets, based on actions of a certain simplicial monoid $E\mathcal M$originally appearing in the construction of global algebraic $K$-theory. As ourmain results, we show that strictly commutative monoids with respect to acertain box product on these simplicial $*$-modules yield models ofequivariantly and globally coherently commutative monoids, and we give acharacterization of simplicial $*$-modules in terms of a certain mildnesscondition on the $E\mathcal M$-action, relaxing the notion of tamenesspreviously investigated by Sagave-Schwede and the first author.