  We study the boundary of the Milnor fibre of real analytic singularities $f:
(\bR^m,0) \to (\bR^k,0)$, $m\geq k$, with an isolated critical value and the
Thom $a_f$-property. We define the vanishing zone for $f$ and we give necessary
and sufficient conditions for it to be a fibre bundle over the link of the
singular set of $f^{-1}(0)$. In the case of singularities of the type $\fgbar:
(\bC^n,0) \to (\bC,0)$ with an isoalted critical value, $f, g$ holomorphic, we
further describe the degeneration of the boundary of the Milnor fibre to the
link of $\fgbar$. As a milestone, we also construct a L\^e's polyhedron for
real analytic singularities of the type $\fgbar: (\bC^2,0) \to (\bC,0)$ such
that either $f$ or $g$ depends only on one variable.
