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