The realization space is
  [1   0   1   0   1    0                                                                                           x3^2                                                                                                        x3^2 - x3                                                                                           x3^2                                  x3    1]
  [0   1   1   0   0    1                                                                                           x3^2   x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 3*x1*x3^2 + x1*x3 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - x3^2                                                       x1*x2*x3^2 + x1*x2*x3 - 2*x1*x3^2 + x3^3   x1*x2*x3 + x1*x2 - 2*x1*x3 + x3^2   x2]
  [0   0   0   1   1   -1   x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - 2*x3^2 + x3                                                                                                  x1*x3^2 - x1*x3   x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - 2*x3^2 + x3                                x3^2   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 8 generators
avoiding the zero loci of the polynomials
RingElem[x1*x2*x3 + x1*x2 - 2*x1*x3 - x2*x3 + x3^2, x3, x3 - 1, x2 - 1, x1*x2*x3 + x1*x2 - 2*x1*x3 + 2*x3^2 - x3, x1 - x3, x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 - 4*x1*x2*x3 - x1*x2 + 2*x1*x3^2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 - x3^3 - x3^2 + x3, x2 + x3 - 1, x1*x2^2*x3 + x1*x2^2 - x1*x2*x3^2 - 3*x1*x2*x3 + 2*x1*x3^2 + 2*x2*x3^2 - x2*x3 - x3^3, x1*x2*x3 + x1*x2 - 2*x1*x3 + x3^2 - x3, x1^2*x2^4*x3^2 + 2*x1^2*x2^4*x3 + x1^2*x2^4 + x1^2*x2^3*x3^3 - 4*x1^2*x2^3*x3^2 - 7*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 6*x1^2*x2^2*x3^3 + 5*x1^2*x2^2*x3^2 + 8*x1^2*x2^2*x3 + x1^2*x2^2 - x1^2*x2*x3^4 + 11*x1^2*x2*x3^3 - 3*x1^2*x2*x3^2 - 3*x1^2*x2*x3 + 2*x1^2*x3^4 - 8*x1^2*x3^3 + 2*x1^2*x3^2 + 3*x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 - x1*x2^3*x3 + 4*x1*x2^2*x3^4 - 9*x1*x2^2*x3^3 - 3*x1*x2^2*x3^2 + 2*x1*x2^2*x3 + x1*x2*x3^5 - 13*x1*x2*x3^4 + 11*x1*x2*x3^3 - x1*x2*x3 - 3*x1*x3^5 + 13*x1*x3^4 - 7*x1*x3^3 + x1*x3^2 + 2*x2^2*x3^4 - x2^2*x3^3 + 3*x2*x3^5 - 5*x2*x3^4 + 2*x2*x3^3 + x3^6 - 5*x3^5 + 4*x3^4 - x3^3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 3*x1*x2*x3 - 3*x1*x3^2 + x1*x3 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - x3^2, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 3*x1*x3^2 + x1*x3 + x2*x3^2 + 2*x3^3 - x3^2, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 3*x1*x2*x3 - x1*x2 - 3*x1*x3^2 + x1*x3 + 2*x2*x3^2 + 2*x3^3 - x3^2, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 2*x1*x3^2 + x2*x3^2 + x3^3, x1^2*x2^3*x3^2 + 2*x1^2*x2^3*x3 + x1^2*x2^3 + x1^2*x2^2*x3^3 - 3*x1^2*x2^2*x3^2 - 5*x1^2*x2^2*x3 - x1^2*x2^2 - 5*x1^2*x2*x3^3 + 2*x1^2*x2*x3^2 + 3*x1^2*x2*x3 + 6*x1^2*x3^3 - 2*x1^2*x3^2 + 3*x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - x1*x2^2*x3 + 4*x1*x2*x3^4 - 6*x1*x2*x3^3 - x1*x2*x3^2 + x1*x2*x3 - 9*x1*x3^4 + 6*x1*x3^3 - x1*x3^2 + 2*x2*x3^4 - x2*x3^3 + 3*x3^5 - 3*x3^4 + x3^3, x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 + 2*x2*x3^2 - x2*x3 + x3^3, x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - 2*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - x3^2, x2 + x3, x2, x1*x2*x3 + x1*x2 - 2*x1*x3 + x3^2, x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 - x1*x3^2 - x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3, x1^2*x2*x3^2 - x1^2*x2 - 2*x1^2*x3^2 + 2*x1^2*x3 - x1*x2^2*x3^2 - x1*x2^2*x3 - x1*x2*x3^3 + 2*x1*x2*x3^2 + x1*x2*x3 + 4*x1*x3^3 - 2*x1*x3^2 - 2*x2*x3^3 + x2*x3^2 - 2*x3^4 + x3^3, x1^2*x2*x3^2 - x1^2*x2 - 2*x1^2*x3^2 + 2*x1^2*x3 - x1*x2^2*x3^2 - x1*x2^2*x3 - x1*x2*x3^3 + 2*x1*x2*x3^2 + x1*x2*x3 + 4*x1*x3^3 - 3*x1*x3^2 + x1*x3 - 2*x2*x3^3 + x2*x3^2 - 2*x3^4 + 2*x3^3 - x3^2, x1^2*x2*x3 + x1^2*x2 - 2*x1^2*x3 - x1*x2^2*x3 - x1*x2^2 - x1*x2*x3^2 + x1*x2*x3 + 4*x1*x3^2 + x1*x3 - 2*x2*x3^2 + x2*x3 - 2*x3^3, x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3 - 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^2, x1^2*x2^4*x3^2 + 2*x1^2*x2^4*x3 + x1^2*x2^4 - 5*x1^2*x2^3*x3^2 - 6*x1^2*x2^3*x3 - x1^2*x2^3 - x1^2*x2^2*x3^3 + 6*x1^2*x2^2*x3^2 + 3*x1^2*x2^2*x3 + x1^2*x2*x3^4 + 3*x1^2*x2*x3^3 - x1^2*x2*x3^2 + x1^2*x2*x3 - 2*x1^2*x3^4 - 2*x1^2*x3^2 + 4*x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 - 2*x1*x2^3*x3 + x1*x2^2*x3^4 - 11*x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 + 2*x1*x2^2*x3 - x1*x2*x3^5 - 5*x1*x2*x3^4 + 7*x1*x2*x3^3 - 3*x1*x2*x3^2 + 3*x1*x3^5 + 2*x1*x3^3 - x1*x3^2 + 4*x2^2*x3^4 - 4*x2^2*x3^3 + x2^2*x3^2 + 2*x2*x3^5 - 5*x2*x3^4 + 4*x2*x3^3 - x2*x3^2 - x3^6, x1^2*x2^3*x3^2 + 2*x1^2*x2^3*x3 + x1^2*x2^3 - 5*x1^2*x2^2*x3^2 - 6*x1^2*x2^2*x3 - x1^2*x2^2 + 8*x1^2*x2*x3^2 + 4*x1^2*x2*x3 - 4*x1^2*x3^2 + 3*x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - x1*x2^2*x3 + x1*x2*x3^4 - 8*x1*x2*x3^3 + x1*x2*x3 - 2*x1*x3^4 + 6*x1*x3^3 - 2*x1*x3^2 + 2*x2*x3^4 - x2*x3^3 + x3^5 - 3*x3^4 + x3^3, x1*x2*x3 + x1*x2 - 2*x1*x3 + 2*x3^2, x1^2*x2^4*x3^2 + 2*x1^2*x2^4*x3 + x1^2*x2^4 + x1^2*x2^3*x3^3 - 4*x1^2*x2^3*x3^2 - 7*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 6*x1^2*x2^2*x3^3 + 5*x1^2*x2^2*x3^2 + 8*x1^2*x2^2*x3 + x1^2*x2^2 - x1^2*x2*x3^4 + 11*x1^2*x2*x3^3 - 3*x1^2*x2*x3^2 - 3*x1^2*x2*x3 + 2*x1^2*x3^4 - 8*x1^2*x3^3 + 2*x1^2*x3^2 + 4*x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 - 2*x1*x2^3*x3 + 5*x1*x2^2*x3^4 - 11*x1*x2^2*x3^3 - x1*x2^2*x3^2 + 3*x1*x2^2*x3 + x1*x2*x3^5 - 16*x1*x2*x3^4 + 14*x1*x2*x3^3 - 2*x1*x2*x3^2 - x1*x2*x3 - 4*x1*x3^5 + 12*x1*x3^4 - 7*x1*x3^3 + x1*x3^2 + 4*x2^2*x3^4 - 4*x2^2*x3^3 + x2^2*x3^2 + 6*x2*x3^5 - 9*x2*x3^4 + 5*x2*x3^3 - x2*x3^2 + 2*x3^6 - 5*x3^5 + 4*x3^4 - x3^3, x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 - x1*x3^2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - 2*x3^2 + x3, x1^2*x2^4*x3^2 + 2*x1^2*x2^4*x3 + x1^2*x2^4 + x1^2*x2^3*x3^3 - 4*x1^2*x2^3*x3^2 - 7*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 6*x1^2*x2^2*x3^3 + 5*x1^2*x2^2*x3^2 + 8*x1^2*x2^2*x3 + x1^2*x2^2 - x1^2*x2*x3^4 + 11*x1^2*x2*x3^3 - 3*x1^2*x2*x3^2 - 3*x1^2*x2*x3 + 2*x1^2*x3^4 - 8*x1^2*x3^3 + 2*x1^2*x3^2 + 4*x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 - 2*x1*x2^3*x3 + 5*x1*x2^2*x3^4 - 12*x1*x2^2*x3^3 - x1*x2^2*x3^2 + 4*x1*x2^2*x3 + x1*x2*x3^5 - 16*x1*x2*x3^4 + 17*x1*x2*x3^3 - 4*x1*x2*x3^2 - 2*x1*x2*x3 - 4*x1*x3^5 + 13*x1*x3^4 - 10*x1*x3^3 + 3*x1*x3^2 + 4*x2^2*x3^4 - 4*x2^2*x3^3 + x2^2*x3^2 + 6*x2*x3^5 - 11*x2*x3^4 + 8*x2*x3^3 - 2*x2*x3^2 + 2*x3^6 - 6*x3^5 + 7*x3^4 - 4*x3^3 + x3^2, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 4*x1*x3^2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + 3*x3^3 - 3*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - 3*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 2*x1*x3^2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - 2*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 3*x1*x3^2 + x1*x3 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - 2*x3^2 + x3, x1^2*x2^4*x3^2 + 2*x1^2*x2^4*x3 + x1^2*x2^4 + x1^2*x2^3*x3^3 - 4*x1^2*x2^3*x3^2 - 7*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 6*x1^2*x2^2*x3^3 + 5*x1^2*x2^2*x3^2 + 8*x1^2*x2^2*x3 + x1^2*x2^2 + 11*x1^2*x2*x3^3 - 4*x1^2*x2*x3^2 - 3*x1^2*x2*x3 - 6*x1^2*x3^3 + 2*x1^2*x3^2 + 4*x1*x2^3*x3^3 + 2*x1*x2^3*x3^2 - 2*x1*x2^3*x3 + 5*x1*x2^2*x3^4 - 12*x1*x2^2*x3^3 - 2*x1*x2^2*x3^2 + 3*x1*x2^2*x3 + x1*x2*x3^5 - 17*x1*x2*x3^4 + 16*x1*x2*x3^3 - x1*x2*x3^2 - x1*x2*x3 - 3*x1*x3^5 + 13*x1*x3^4 - 7*x1*x3^3 + x1*x3^2 + 4*x2^2*x3^4 - 4*x2^2*x3^3 + x2^2*x3^2 + 6*x2*x3^5 - 11*x2*x3^4 + 6*x2*x3^3 - x2*x3^2 + 2*x3^6 - 7*x3^5 + 6*x3^4 - 2*x3^3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^3 + x1*x2*x3^2 - 3*x1*x2*x3 - x1*x2 - 2*x1*x3^3 - 2*x1*x3^2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^4 + x3^3 - 2*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 2*x1*x3^2 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - x3^2, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 2*x1*x3^2 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - 2*x3^2 + x3, x1*x2^2*x3 + x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x2 - 3*x1*x3^2 + x1*x3 + 2*x2*x3^2 - x2*x3 + 2*x3^3 - x3^2, x1 - 1, x1, x1*x2^2*x3 + x1*x2^2 - 3*x1*x2*x3 - x1*x2 + 2*x1*x3 + 2*x2*x3^2 - x2*x3 + x3^3 - x3^2 + x3]