  We propose an approach for the computation of multi-parameter families of
Galois extensions with prescribed ramification type. More precisely, we combine
existing deformation and interpolation techniques with recently developed
strong tools for the computation of $3$-point covers. To demonstrate the
applicability of our method in relatively large degrees, we compute several
families of polynomials with symplectic Galois groups, in particular obtaining
the first totally real polynomials with Galois group PSp(6,2).
