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