  In this paper, we introduce a new regularity condition that characterizes the
tameness of a composite singularity $H=G\circ F$ in a sharp way. Our approach
provides a natural tool that links the topology of the Milnor tube fibrations
through the Milnor fibers of the respective components of the map germs $F$,
$G$ and $H = G\circ F$. We also study the invariance of tameness by
$\mathcal{L}$-equivalence, $\mathcal{R}$-equivalence, and hence by
$\mathcal{A}$-equivalence, and we give conditions for when two component map
germs of the composite singularity $H=G\circ F$ being tame implies the third
one is tame. As an application, we show how to relate the Euler characteristics
of the Milnor fibers of $F,G$ and $H$ to each other.
