  For each pair $(m,n)$ of positive integers with $(m,n)\not= (1,1)$ and an
arbitrary field $\bf F$ with algebraic closure $\overline{\bf F}$, let $\rm
Po^{d,m}_n(\bf F)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in
\bf F [z]^m$ of $\bf F$-coefficients monic polynomials of the same degree $d$
such that the polynomials $\{f_k(z)\}_{k=1}^m$ have no common root in
$\overline{\bf F}$ of multiplicity $\geq n$. These spaces $\rm Po^{d,m}_n(\bf
F)$ were first defined and studied by B. Farb and J. Wolfson as generalizations
of spaces first studied by Arnold, Vassiliev and Segal and others in several
different contexts. In previous we determined explicitly the homotopy type of
this space in the case $\bf F =\Bbb C$. In this paper, we investigate the case
$\bf F =\Bbb R$.
