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