The realization space is
  [1   1   0   1   0    0                x1*x3                                     x1*x3      x3                x1*x3    1]
  [0   1   1   0   0    1   x1*x2 + x1*x3 - x3                                   x1^2*x2   x1*x2   x1*x2 + x1*x3 - x3   x2]
  [0   0   0   1   1   -1                x1*x3   -x1^2*x2 + x1^2*x3 + x1*x2 + x1*x3 - x3   x1*x3              x1^2*x3   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*x3 - x1^2*x3^2 - 2*x1*x2*x3 - x1*x3^2 + x3^2)
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x3, x1 - 1, x1^2*x2 - x1^2 - x1*x2 + x3, x1 - x3, x1^2*x2*x3 - 2*x1*x2*x3 + x1*x2 - x1*x3^2 + x1*x3 + x3^2 - x3, x1*x2*x3 - x1*x2 - x1*x3 + x3, x1^2*x3 - x1*x2*x3 + x1*x2 - x1*x3^2 + x1*x3 - x3, x2, x2 + x3, x3, x1^2*x2^2 - x1^2*x2 + x1^2*x3 - x1*x2^2 - x1*x2*x3 + x1*x2 - x1*x3^2 + x1*x3 + x2*x3 - x3, x1^2*x2 - x1^2*x3 - x1*x2 + x1*x3^2 - x1*x3 + x3, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x3^2 - x1*x2*x3^2 + 2*x1*x2*x3 - x1*x3^3 + x1*x3^2 - x3^2, x3 - 1, x1*x2 + x1*x3 - x3, x1*x2*x3 - x1*x2 + x1*x3^2 - 2*x1*x3 + x3, x2 + x3 - 1, x2 - 1, x1, x1^2*x3 + x1*x2 + x1*x3 - x3, x1^2*x3 + x1*x2 - x3, x1*x2 - x1*x3 - x3, x1^2*x2 - x1^2*x3 - x1*x2 - x1*x3 + x3, x1*x2 + 2*x1*x3 - x3]