Singular VS singular-factory

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).

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)