The realization space is
  [1                   x2*x3   1   1   0      x2*x3 - x3   0   x2                   x2*x3    0    1]
  [1   -x1*x2^2 + x2*x3 + x3   0   0   1   -x1*x2^2 + x3   0   x2   -x1*x2^2 + x2*x3 + x3    1   x2]
  [1   -x1*x2^2 + x2*x3 + x3   0   1   0               0   1   x3                    x3^2   x1   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal (x1^2*x2^2 - x1*x2*x3 - x1*x3 + x3^2)
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x1 - x3 + 1, x1^2*x2^2 + x1*x2^2*x3 - x1*x2^2 - x1*x2*x3 - x1*x3 - x2*x3^2 + x2*x3 + x3, x1*x2 - x3, x1*x2 - x3 + 1, x1, x1^2*x2^2 + x1*x2^2*x3 - x1*x2*x3 - x1*x3 - x2*x3^2 + x3^2, x2 - 1, x1*x2^2*x3 - x1*x2^2 - x2^2*x3 + x2*x3^2 + x2*x3 - 2*x3^2 + x3, x1*x2^2 - x2*x3 + x3^2 - x3, x1*x2^2 + x2^2*x3 - x2*x3 - x3, x1*x2^2 - x3, x1*x2^2*x3 - x1*x2^2 - x2^2*x3 + x2*x3 - x3^2 + x3, x3, x2 - x3, x1*x2^2 - x2*x3 - x3, x2, x3 - 1, x1*x2^2 + x2*x3^2 - x2*x3 - x3, x1*x2^3 - x1*x2^2*x3 + x1*x2^2 + x2*x3^2 - 2*x2*x3 + x3^2 - x3, x1^2*x2^2 - x1*x3 - x2*x3 + x3^2, x1^2*x2^2 - x1*x2*x3 - x1*x3 - x2*x3 + x3^2, x1^2*x2^2 - x1*x2^2 - x1*x3 + x2*x3 - x3^2 + x3, x1*x2 + x2 - x3, x1^2*x2^3 + x1*x2^2*x3 - 2*x1*x2*x3 - x2*x3^2 + x3^2, x1^2*x2^2 + x1*x2*x3 - 2*x1*x3 - x2*x3 + x3, x1^2*x2^2 + x1*x2^3 + x1*x2^2*x3 - x1*x2^2 - x1*x2*x3 - x1*x3 - x2^2*x3 + x3, x1^2*x2^2 - x1*x3 - x2*x3 + x3, x1^2*x2^2 - x1*x2^2 - x1*x2*x3 - x1*x3 + x3, x1 - 1, x1*x2^2 + x2*x3 - 2*x3, x1^2*x2^5 - 3*x1*x2^3*x3 + x1*x2^2*x3^2 - x2^3*x3^2 + x2^2*x3^3 + x2^2*x3^2 - x2*x3^3 + 2*x2*x3^2 - x3^3, x1*x2^3 - x1*x2^2*x3 + x2^2*x3 - 2*x2*x3 + x3^2, 2*x1*x2^3 - x1*x2^2*x3 - x2^2*x3 + x2*x3^2 - 2*x2*x3 + x3^2]