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