Let \(F:(\mathbb {R}^M,0)\rightarrow (\mathbb {R}^N,0)\) and \(G:(\mathbb {R}^N,0) \rightarrow (\mathbb {R}^K,0)\) , \(M \ge N \ge K \ge 2\) , be non-constant real analytic map germs with isolated critical values. In this paper we study the topology of the Milnor tube fibrations of the map germs F, G and their composition \(H = G\circ F\) , under the tame condition. More precisely, we show that the Milnor fiber \(\mathcal {F}_H\) of H is homotopy equivalent to the product \(\mathcal {F}_F \times \mathcal {F}_G\) of the Milnor fibers of F and G, and that the boundary \(\partial \mathcal {F}_H\) of \(\mathcal {F}_H\) is homotopy equivalent to \(\partial (\mathcal {F}_F \times \mathcal {F}_G) = (\partial \mathcal {F}_F \times \mathcal {F}_G) \cup (\mathcal {F}_F \times \partial \mathcal {F}_G)\) . Furthermore, if each component of \(\partial (\mathcal {F}_F \times \mathcal {F}_G)\) is simply connected and \(M - K \ge 6\) , then the homotopy equivalences can be replaced by diffeomorphisms in the above statements.
