We study the topology of the boundaries \(\partial F_{f}\) and \(\partial F_{I}\) of the Milnor fibers \(F_{f}\) and \(F_{I},\) respectively, of real analytic map-germs \(f{:}\,(\mathbb {R}^M,0) \rightarrow (\mathbb {R}^K,0)\) and \(f_{I}:=\Pi _{I}\circ f{:}\,(\mathbb {R}^M,0) \rightarrow (\mathbb {R}^I,0)\) that admit Milnor’s tube fibrations, where \(\Pi _{I}{:}\,({\mathbb {R}}^K,0)\rightarrow ({\mathbb {R}}^{I},0)\) is the canonical projection for \(1\le I<K.\) For each I we prove that the Milnor boundary \(\partial F_{I}\) is given by the double of the Milnor tube fiber \(F_{I+1}.\) Beside that, if \(K-I\ge 2,\) we prove that the pair \((\partial F_{I},\partial F_{f})\) is a generalized \((K-I-1)\) -open-book decomposition with binding \(\partial F_{f}\) and page \(F_{f}{\setminus }\partial F_{f}\) —the interior of the Milnor fibre \(F_{f}.\) This allows us to prove several new Euler characteristic formulae connecting the Milnor boundaries \(\partial F_{f},\) \(\partial F_{I},\) with the respective links \(\mathcal {L}_{f}, \mathcal {L}_{I},\) for each \(1\le I<K,\) and a Lê–Greuel type formula for the Milnor boundary.
