singular-factory VS Singular

Your best bet would be just call out to, say, Singular; either directly executing it by supplying textual input, or make a Haskell binding for the relevant parts (for example I wrote Haskell bindings to the multivariate polynomial factorization engine. It was kind of painful)

As for the "reasonably straightforward" comment, I was looking at Singular's triangMH function which does most of the work in solving such a polynomial system, and it didn't seem too tricky. It starts off with a reduced lexicographic Gröbner basis and eliminates variables one at a time by recursively calling itself, yielding a triangular decomposition of the 0-dimensional variety. Then a univariate root finding algorithm can be used to solve for each variable in turn. It might not have the best numerical stability properties or etc, but it seems manageable (given a basic framework for commutative algebra). I agree that, in general, it's unreasonable to expect to be able to cook up something in a few days of work that is of equivalent robustness to these battle-tested algorithms (for instance I can see all the complexity that's in Singular's Laguerre solver to attempt to improve numerical stability, compared to the simple Laguerre solver I wrote myself). However this case didn't seem quite as difficult as usual (for Singular's algorithm specifically, PHCpack's algorithm looks extremely involved).