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