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