  We consider systems of polynomial equations and inequalities in
$\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1,
\ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$
indeterminates are considered as parameters and we assume that when
specialising them generically, the set of common complex solutions, to the
obtained equations, is finite. We consider the problem of real root
classification for such parameter-dependent problems, i.e. identifying the
possible number of real solutions depending on the values of the parameters and
computing a description of the regions of the space of parameters over which
the number of real roots remains invariant.
  We design an algorithm for solving this problem. The formulas it outputs
enjoy a determinantal structure. Under genericity assumptions, we show that its
arithmetic complexity is polynomial in both the maximum degree $d$ and the
number $s$ of the input inequalities and exponential in $nt+t^2$. The output
formulas consist of polynomials of degree bounded by $(2s+n)d^{n+1}$. This is
the first algorithm with such a singly exponential complexity. We report on
practical experiments showing that a first implementation of this algorithm can
tackle examples which were previously out of reach.
