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