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