The realization space is
  [1   0   1   0   1    0                                                           x2^3 + x2^2*x3 - x2^2                                                       x2^2*x3 + x2*x3^2 - x2*x3                                                                    x2^3*x3 + x2^2*x3^2 - x2^2*x3                x3    1]
  [0   1   1   0   0    1                                                           x2^3 + x2^2*x3 - x2^2   -x1^2*x3 + x1*x2^3 - 2*x1*x2^2 + x1*x2*x3 + x1*x2 + x1*x3^2 + x2^2*x3 - x2*x3        -x1^2*x2*x3 + x1*x2^4 - 2*x1*x2^3 + x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 + x2^3*x3 - x2^2*x3   x1*x2 - x1 + x3   x2]
  [0   0   0   1   1   -1   -x1^2*x3 + x1*x2^3 - 2*x1*x2^2 + x1*x2*x3 + x1*x2 + x1*x3^2 + x2^2*x3 - x2*x3                                              x1*x2^2*x3 + x1*x2*x3^2 - x1*x2*x3   -x1^2*x3^2 + x1*x2^3*x3 - 2*x1*x2^2*x3 + x1*x2*x3^2 + x1*x2*x3 + x1*x3^3 + x2^2*x3^2 - 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 - 1, x2 + x3 - 1, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 - x2^2*x3 + x2*x3, x1^2 + x1*x2^2 - 2*x1*x2 - x1*x3 + x2*x3, x3, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 + x2^3*x3 + x2^2*x3^2 - 2*x2^2*x3 + x2*x3, x2, x2 + x3, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 + x2*x3^2, x1^2*x3 + x1*x2^2*x3 + x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 - x2^2*x3 + x2*x3, x1^2*x3^2 - x1^2*x3 + x1*x2^3 + x1*x2^2*x3^2 + x1*x2^2*x3 - 2*x1*x2^2 - x1*x2*x3^2 + x1*x2 - x1*x3^3 + x1*x3^2 - x2^3*x3 - 2*x2^2*x3^2 + 2*x2^2*x3 + x2*x3^2 - x2*x3, x1^2*x3 - x1*x2^3 - x1*x2^2*x3 + 2*x1*x2^2 - x1*x2*x3^2 - x1*x2 - x1*x3^2 + x2^3*x3 + 2*x2^2*x3^2 - 2*x2^2*x3 + x2*x3^3 - x2*x3^2 + x2*x3, x1^4*x3^2 - 2*x1^3*x2^3*x3 + 4*x1^3*x2^2*x3 - 2*x1^3*x2*x3^2 - 2*x1^3*x2*x3 - 2*x1^3*x3^3 + x1^2*x2^6 - 4*x1^2*x2^5 + 2*x1^2*x2^4*x3 + 6*x1^2*x2^4 + 4*x1^2*x2^3*x3^2 - 4*x1^2*x2^3*x3 - 4*x1^2*x2^3 + 2*x1^2*x2^2*x3^3 - 7*x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 + x1^2*x2^2 + 2*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 + x1^2*x3^4 - x1*x2^6*x3 + 5*x1*x2^5*x3 + x1*x2^4*x3^3 - 2*x1*x2^4*x3^2 - 9*x1*x2^4*x3 - 5*x1*x2^3*x3^3 + 4*x1*x2^3*x3^2 + 7*x1*x2^3*x3 - 2*x1*x2^2*x3^4 + 4*x1*x2^2*x3^3 - 2*x1*x2^2*x3^2 - 2*x1*x2^2*x3 - 2*x1*x2*x3^3 - x2^5*x3^2 + 3*x2^4*x3^2 + x2^3*x3^4 - 3*x2^3*x3^2 + x2^2*x3^2, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 + x2^3 + x2^2*x3 - 2*x2^2 + x2*x3^2 - x2*x3 + x2, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 + x2^4 + 2*x2^3*x3 - 2*x2^3 + x2^2*x3^2 - 3*x2^2*x3 + x2^2 + x2*x3, x3 - 1, x1, x1^2*x2 + x1^2*x3 + x1*x2^3 + x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x2*x3, x1 - 1, x1^3*x2*x3 - x1^3*x3 - x1^2*x2^4 + 3*x1^2*x2^3 - x1^2*x2^2*x3 - 3*x1^2*x2^2 - x1^2*x2*x3^2 + x1^2*x2*x3 + x1^2*x2 + 2*x1^2*x3^2 - x1*x2^3*x3 + x1*x2^2*x3^2 + 3*x1*x2^2*x3 - x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 - x2^2*x3^2 + x2*x3^2, x1^3*x2*x3 - x1^3*x3 - x1^2*x2^4 + 3*x1^2*x2^3 - x1^2*x2^2*x3 - 3*x1^2*x2^2 - x1^2*x2*x3^2 + x1^2*x2*x3 + x1^2*x2 + 2*x1^2*x3^2 + x1*x2^4 - 2*x1*x2^3 + x1*x2^2*x3^2 + 2*x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 - x2^2*x3^2 + x2*x3^2, x1^2*x3^2 + x1*x2^4 + x1*x2^3*x3 - 2*x1*x2^3 + x1*x2^2*x3^2 + x1*x2^2 - x1*x2*x3^2 - x1*x2*x3 - x1*x3^3 - x2^2*x3^2 + x2*x3^2, x1*x2 + x1*x3 - x1 + x3, x1*x2 - x1 + x3, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 + x2^3 - x2^2 + x2*x3, x1^2*x2*x3 + x1^2*x3^2 - x1*x2^4 - x1*x2^3*x3 + 2*x1*x2^3 + x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3^2 - x1*x2*x3 - x1*x3^3 + x2*x3^2, x1^4*x3^2 - 2*x1^3*x2^3*x3 + 4*x1^3*x2^2*x3 - 2*x1^3*x2*x3^2 - 2*x1^3*x2*x3 - 2*x1^3*x3^3 + x1^2*x2^6 - 4*x1^2*x2^5 + 2*x1^2*x2^4*x3 + 6*x1^2*x2^4 + 2*x1^2*x2^3*x3^2 - 3*x1^2*x2^3*x3 - 4*x1^2*x2^3 - 4*x1^2*x2^2*x3^2 + x1^2*x2^2*x3 + x1^2*x2^2 + 2*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 + x1^2*x3^4 - x1*x2^6 + 3*x1*x2^5 - 2*x1*x2^4*x3^2 - 3*x1*x2^4*x3 - 3*x1*x2^4 - x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 + 5*x1*x2^3*x3 + x1*x2^3 + x1*x2^2*x3^3 - x1*x2^2*x3^2 - 2*x1*x2^2*x3 - 2*x1*x2*x3^3 + 2*x2^4*x3^2 + x2^3*x3^3 - 3*x2^3*x3^2 + x2^2*x3^2, x1^2*x2*x3 - x1^2*x3^2 - x1*x2^4 + 2*x1*x2^3 - x1*x2^2*x3^2 - 2*x1*x2^2*x3 - x1*x2^2 + x1*x2*x3 + x1*x3^3 + 2*x2^2*x3^2 - x2*x3^2, x1^2*x3 - x1*x2^3 - x1*x2^2*x3 + 2*x1*x2^2 - x1*x2*x3^2 - x1*x2 - x1*x3^2 - x2^2*x3 + x2*x3, x1^2*x3 - x1*x2^3 - x1*x2^2*x3 + 2*x1*x2^2 - x1*x2*x3^2 - x1*x2 - x1*x3^2 + x2*x3^2, x1^2*x3 - x1*x2^3 + 2*x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3^2 - x2^3 - 2*x2^2*x3 + x2^2 + x2*x3]