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