The realization space is
  [1   1   0   0   1   1                                                                0                                            x2*x3^3 - 2*x2*x3^2 + x2*x3 - x3^4 + 2*x3^3 - x3^2                                                                        x2*x3^3 - 2*x2*x3^2 + x2*x3 - x3^4 + 2*x3^3 - x3^2      x3    1]
  [0   1   1   0   0   1                                    x2*x3^2 - x2*x3 - x3^3 + x3^2   x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3 - x1*x2*x3^3 + x1*x2*x3 + x2^2*x3 + x2*x3^3 - x2*x3^2                               x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3 - x1*x2*x3^3 + x1*x2*x3 + x2^2*x3 + x2*x3^3 - x2*x3^2   x1*x2   x2]
  [0   0   0   1   1   1   -x1^2*x2*x3 + x1^2*x2 + x1*x2*x3^2 + x1*x3^2 - x1*x3 - x2*x3^2                          x1*x2*x3^3 - 2*x1*x2*x3^2 + x1*x2*x3 - x1*x3^4 + 2*x1*x3^3 - x1*x3^2   x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x2 - x1*x2*x3^2 + x1*x2*x3 - x1*x3^4 + x1*x3^3 + x1*x3^2 - x1*x3 + x2*x3^3 - x2*x3^2   x1*x3   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 1 generator
avoiding the zero loci of the polynomials
RingElem[x1 - x3, x2, x2 - x3, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3^2 - x1*x2*x3^2 + x1*x2*x3 + x2^2*x3^2 + x2*x3^3 - x2*x3^2 - x3^4 + x3^3, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3^2 + x1*x2*x3 - x1*x2 - 2*x1*x3^2 + 2*x1*x3 + x2*x3 + x3^3 - x3^2, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3 - x1*x3^3 + x1*x3 - x2*x3^3 + 2*x2*x3^2 + x3^4 - x3^3, x2 + x3 - 1, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3^2 - x1*x2*x3^2 + x1*x2*x3 + x2^2*x3^2 + x2*x3^3 - 2*x2*x3^2 + x2*x3 - x3^4 + 2*x3^3 - x3^2, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2*x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3^2 + x1*x2^2*x3 - x1*x2^2 - x1*x2*x3^2 + x1*x2 - x1*x3^3 + 2*x1*x3^2 - x1*x3 + x2^2*x3 + x2*x3^2 - x2*x3 + x3^4 - 2*x3^3 + x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3^2 - x1*x3^3 + x1*x3^2 + x2^2*x3^2 - x2*x3^3 + x2*x3^2 + x3^4 - x3^3, x1^4*x2^3*x3^2 - 2*x1^4*x2^3*x3 + x1^4*x2^3 - x1^3*x2^3*x3^3 + x1^3*x2^3*x3 - x1^3*x2^2*x3^4 + 3*x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 - x1^2*x2^3*x3^4 + 5*x1^2*x2^3*x3^3 - 3*x1^2*x2^3*x3^2 + 2*x1^2*x2^2*x3^5 - 2*x1^2*x2^2*x3^4 + x1^2*x2^2*x3^3 - x1^2*x2^2*x3^2 + x1^2*x2*x3^5 - x1^2*x2*x3^4 - x1^2*x2*x3^3 + x1^2*x2*x3^2 + x1*x2^3*x3^5 - 2*x1*x2^3*x3^4 - x1*x2^3*x3^3 - x1*x2^2*x3^6 + 2*x1*x2^2*x3^5 - 6*x1*x2^2*x3^4 + 6*x1*x2^2*x3^3 - x1*x2^2*x3^2 - 3*x1*x2*x3^6 + 8*x1*x2*x3^5 - 7*x1*x2*x3^4 + 2*x1*x2*x3^3 + x1*x3^7 - 3*x1*x3^6 + 3*x1*x3^5 - x1*x3^4 - x2^3*x3^5 + 2*x2^3*x3^4 + 2*x2^2*x3^5 - 3*x2^2*x3^4 + x2^2*x3^3 + 2*x2*x3^7 - 6*x2*x3^6 + 6*x2*x3^5 - 2*x2*x3^4 - x3^8 + 3*x3^7 - 3*x3^6 + x3^5, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - x1*x2^2*x3^2 + x1*x2*x3 + x1*x3^2 - x1*x3 + x2^2*x3^2 + x2*x3^3 - 2*x2*x3^2 - x3^4 + x3^3, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - x1*x2^2*x3^2 + x1*x2*x3 + x1*x3^2 - x1*x3 + x2^2*x3^2 + x2*x3^3 - 3*x2*x3^2 + x2*x3 - x3^4 + 2*x3^3 - x3^2, x2 - 1, x3 - 1, x3, x1, x1^3*x2^2*x3 - x1^3*x2^2 - x1^2*x2^2*x3 - 2*x1^2*x2*x3^2 + 2*x1^2*x2*x3 - x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - x1*x3^4 + 2*x1*x3^3 - x1*x3^2 + x2*x3^4 - x2*x3^3, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3 - x1*x3^3 + x1*x3 + x2*x3 + x3^3 - x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3^2 + x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3^2 + 2*x1*x2*x3 + x2^2*x3 + 2*x2*x3^3 - 3*x2*x3^2 + x2*x3 - x3^4 + 2*x3^3 - x3^2, x1 - 1, x1^3*x2^2*x3 - x1^3*x2^2 - x1^2*x2^2*x3^2 - 2*x1^2*x2*x3^2 + 2*x1^2*x2*x3 + x1*x2^2*x3^2 + 2*x1*x2*x3^3 - x1*x2*x3^2 - x1*x3^4 + 2*x1*x3^3 - x1*x3^2 - x2*x3^3, x1^3*x2^2*x3 - x1^3*x2^2 - x1^2*x2^2*x3^2 - x1^2*x2*x3^2 + x1^2*x2*x3 + x1*x2^2*x3^2 + x1*x2*x3^3 - x1*x2*x3^2 - x1*x3^4 + x1*x3^3 - x2*x3^3 + x2*x3^2 + x3^4 - x3^3, x1^3*x2^2*x3 - x1^3*x2^2 - x1^2*x2^2*x3^2 - 2*x1^2*x2*x3^2 + 2*x1^2*x2*x3 + x1*x2^2*x3^2 + 2*x1*x2*x3^3 - x1*x2*x3^2 - x1*x3^4 + 2*x1*x3^3 - x1*x3^2 - 2*x2*x3^3 + x2*x3^2 + x3^4 - x3^3, x1*x2 - x3, x1*x2 + x1*x3 - x3, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3 - x1*x2*x3^3 + x1*x2*x3 + x2^2*x3 + x2*x3^2 - x2*x3 + x3^4 - 2*x3^3 + x3^2, x1^4*x2^3*x3^2 - 2*x1^4*x2^3*x3 + x1^4*x2^3 - x1^3*x2^3*x3^3 + x1^3*x2^3*x3 - x1^3*x2^2*x3^4 + 3*x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 + 2*x1^2*x2^3*x3^3 - x1^2*x2^3*x3 + x1^2*x2^2*x3^5 + 2*x1^2*x2^2*x3^4 - 5*x1^2*x2^2*x3^3 + 3*x1^2*x2^2*x3^2 - x1^2*x2^2*x3 + 2*x1^2*x2*x3^4 - 4*x1^2*x2*x3^3 + 2*x1^2*x2*x3^2 - 2*x1*x2^3*x3^3 - 2*x1*x2^2*x3^5 + 2*x1*x2^2*x3^3 - x1*x2*x3^6 + 2*x1*x2*x3^5 - 2*x1*x2*x3^3 + x1*x2*x3^2 + x1*x3^7 - 2*x1*x3^6 + 2*x1*x3^4 - x1*x3^3 + x2^3*x3^3 + 2*x2^2*x3^5 - 3*x2^2*x3^4 + x2^2*x3^3 - x2*x3^6 + 2*x2*x3^5 - x2*x3^4, x1^4*x2^3*x3^2 - 2*x1^4*x2^3*x3 + x1^4*x2^3 - x1^3*x2^3*x3^3 + x1^3*x2^3*x3 - x1^3*x2^2*x3^4 + 3*x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 + 2*x1^2*x2^3*x3^3 - x1^2*x2^3*x3 + x1^2*x2^2*x3^5 + x1^2*x2^2*x3^4 - 2*x1^2*x2^2*x3^3 + x1^2*x2*x3^5 - x1^2*x2*x3^4 - x1^2*x2*x3^3 + x1^2*x2*x3^2 - 2*x1*x2^3*x3^3 - x1*x2^2*x3^5 - 2*x1*x2^2*x3^4 + 3*x1*x2^2*x3^3 - 2*x1*x2*x3^6 + 5*x1*x2*x3^5 - 4*x1*x2*x3^4 + x1*x2*x3^3 + x1*x3^7 - 3*x1*x3^6 + 3*x1*x3^5 - x1*x3^4 + x2^3*x3^3 + 2*x2^2*x3^4 - 3*x2^2*x3^3 + x2^2*x3^2 + 2*x2*x3^6 - 6*x2*x3^5 + 6*x2*x3^4 - 2*x2*x3^3 - x3^7 + 3*x3^6 - 3*x3^5 + x3^4, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3^2 + 2*x1^2*x2*x3 - x1^2*x2 - x1*x2^2*x3 - x1*x2*x3^3 + x1*x2*x3^2 + x1*x3^4 - x1*x3^3 - x1*x3^2 + x1*x3 + x2^2*x3, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3 - x1*x3^3 + x1*x3 + x2*x3^2, x1^4*x2^3*x3^2 - 2*x1^4*x2^3*x3 + x1^4*x2^3 - x1^3*x2^3*x3^3 + x1^3*x2^3*x3 - x1^3*x2^2*x3^4 + 3*x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 + 2*x1^2*x2^3*x3^3 - x1^2*x2^3*x3 + x1^2*x2^2*x3^5 + x1^2*x2^2*x3^3 - 3*x1^2*x2^2*x3^2 + x1^2*x2^2*x3 + 2*x1^2*x2*x3^5 - 4*x1^2*x2*x3^4 + 2*x1^2*x2*x3^3 - 2*x1*x2^3*x3^3 - 4*x1*x2^2*x3^4 + 4*x1*x2^2*x3^3 - 3*x1*x2*x3^6 + 8*x1*x2*x3^5 - 8*x1*x2*x3^4 + 4*x1*x2*x3^3 - x1*x2*x3^2 + x1*x3^7 - 4*x1*x3^6 + 6*x1*x3^5 - 4*x1*x3^4 + x1*x3^3 + x2^3*x3^3 + x2^2*x3^4 - x2^2*x3^3 + x2*x3^6 - 2*x2*x3^5 + x2*x3^4, x1^4*x2^3*x3^2 - 2*x1^4*x2^3*x3 + x1^4*x2^3 - x1^3*x2^3*x3^3 + x1^3*x2^3*x3 - x1^3*x2^2*x3^4 + 3*x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 + 2*x1^2*x2^3*x3^3 - x1^2*x2^3*x3 + x1^2*x2^2*x3^5 + x1^2*x2^2*x3^3 - 3*x1^2*x2^2*x3^2 + x1^2*x2^2*x3 + 2*x1^2*x2*x3^5 - 4*x1^2*x2*x3^4 + 2*x1^2*x2*x3^3 - 2*x1*x2^3*x3^3 - 4*x1*x2^2*x3^4 + 4*x1*x2^2*x3^3 - 3*x1*x2*x3^6 + 8*x1*x2*x3^5 - 8*x1*x2*x3^4 + 4*x1*x2*x3^3 - x1*x2*x3^2 + x1*x3^7 - 4*x1*x3^6 + 6*x1*x3^5 - 4*x1*x3^4 + x1*x3^3 + x2^3*x3^3 - x2^2*x3^5 + 4*x2^2*x3^4 - 4*x2^2*x3^3 + x2^2*x3^2 + 3*x2*x3^6 - 8*x2*x3^5 + 7*x2*x3^4 - 2*x2*x3^3 - x3^7 + 3*x3^6 - 3*x3^5 + x3^4, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3 - 2*x1*x2*x3^3 + 2*x1*x2*x3^2 + x1*x3^4 - 2*x1*x3^3 + x1*x3^2 + x2^2*x3 + x2*x3^3 - x2*x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1*x2^2*x3 - 2*x1*x2*x3^2 + 2*x1*x2*x3 - x1*x3^4 + 2*x1*x3^3 - x1*x3^2 + x2^2*x3 + x2*x3^2 - x2*x3 + x3^4 - 2*x3^3 + x3^2, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + 2*x2*x3^2 - x2*x3 - x3^3 + x3^2, x1^2*x2*x3 - x1^2*x2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2*x3 + x3^3 - x3^2]