  In this paper, we study the topology of real analytic map-germs with isolated
critical value $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We
compare the topology of $f$ with the topology of the compositions $\pi_i^*
\circ f$, where $\pi_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the
projections $(t_1, \dots, t_n) \mapsto (t_1, \dots, t_{i-1}, t_{i+1}, \dots,
t_n)$, for $i=1, \dots, n$. As a main result, we give necessary and sufficient
conditions for $f$ to have a L\^e-Milnor fibration in the tube.
