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The question makes me think of algebras with straightening laws. Algebras generated by minors (Plücker coordinates!) often have combinatorial multiplication rules that might give something like you want. To get started, maybe the example on page 253 of this is helpful: msri.org/people/staff/de/papers/pdfs/1980-001.pdf
The answer depends a lot on the specific question: do you want one solution or all? Is the variety zero dimensional? How do you deal with numerical instability, what is the computational model (complex numbers cannot be stored on an ordinary computer) etc.
