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