  We extend Schwede's work on the unstable global homotopy theory of orthogonal
spaces and $\mathcal{L}$-spaces to the category of $*$-modules (i.e., unstable
$S$-modules). We prove a theorem which transports model structures and their
properties from $\mathcal{L}$-spaces to $*$-modules and show that the resulting
global model structure for $*$-modules is monoidally Quillen equivalent to that
of orthogonal spaces. As a consequence, there are induced Quillen equivalences
between the associated model categories of monoids, which identify equivalent
models for the global homotopy theory of $A_\infty$-spaces.
