The realization space is
  [1                                             x1*x3 - x3   1   0                                      x1*x3 - x3                                             x1*x3 - x3                                             x1*x3 - x3                                      x1*x3 - x3   0    0    1]
  [1   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3 - x3^2   0   1   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3                                       x1*x2*x3 - x2*x3   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3 - x3^2   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3   0    1   x2]
  [1   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3 - x3^2   0   0                                      x1*x3 - x3   x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3 - x3^2                                                      0                    x1^2*x2*x3 - x1*x2*x3 + x3^2   1   x1   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal (x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 + 2*x1*x2*x3 - x3^2)
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x1 - x3 + 1, x2, x1*x2 - x1 - x3, x1 - 1, x1*x2 - x3, x1, x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 + 2*x1*x2*x3 + x1*x3^2 - x1*x3 - x3^2 + x3, x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 - x1*x3^2 + x2*x3, x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 - 2*x1^2*x2*x3 + x1^2*x2 + 3*x1*x2*x3 + x1*x3^2 - 2*x3^2, x2 - 1, x1^2*x2^2 - x1^2*x2 - x1*x2^2 - x1*x2*x3 + x1*x2 + 2*x2*x3 - x3^2, x1^2*x2 - x1^2 - x1*x2 - x1*x3 + x1 + 2*x3, x1^2*x2^2 - x1^2*x2 - x1*x2^2 - x1*x2*x3 + x1*x2 + x1*x3^2 + 2*x2*x3 - 2*x3^2, x1^4*x2^4 - x1^4*x2^3*x3 - 2*x1^4*x2^3 + x1^4*x2^2*x3 + x1^4*x2^2 - 2*x1^3*x2^4 + x1^3*x2^3*x3 + 4*x1^3*x2^3 + x1^3*x2^2*x3^2 - x1^3*x2^2*x3 - 2*x1^3*x2^2 + x1^2*x2^4 + 3*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 5*x1^2*x2^2*x3^2 - 3*x1^2*x2^2*x3 + x1^2*x2^2 + x1^2*x2*x3^3 + 2*x1^2*x2*x3^2 - 3*x1*x2^3*x3 + 3*x1*x2^2*x3^2 + 3*x1*x2^2*x3 + x1*x2*x3^3 - 2*x1*x2*x3^2 - x1*x3^4 + 2*x2^2*x3^2 - 4*x2*x3^3 + 2*x3^4, x1^2*x2 - x1*x2 - x1*x3 + 2*x3, x1^2*x2 - x1*x2 - x1 + x3 + 1, x1^4*x2^3 - x1^4*x2^2 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + 2*x1^3*x2^2 + x1^2*x2^3 + 4*x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3^2 - x1^2*x2*x3 - 3*x1*x2^2*x3 + x1*x2*x3 + x1*x3^3 + 2*x2*x3^2 - 2*x3^3, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - x1*x2^2*x3 + x1*x2^2 + 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x2*x3^2 - 2*x2*x3 - x3^3 + 2*x3^2, x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x2*x3 - x3^2, x3, x1^2*x2^2 - x1^2*x2 - x1*x2^2 - x1*x2*x3 + x1*x2 + 2*x2*x3 + x3^3 - x3^2, x3 - 1, x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 - x1*x3 + x2*x3 + x3, x1^4*x2^4 - 2*x1^4*x2^3 + x1^4*x2^2 - 2*x1^3*x2^4 - x1^3*x2^3*x3 + 4*x1^3*x2^3 - 2*x1^3*x2^2 + x1^3*x2*x3 + x1^2*x2^4 + 4*x1^2*x2^3*x3 - 2*x1^2*x2^3 - x1^2*x2^2*x3^2 - 2*x1^2*x2^2*x3 + x1^2*x2^2 + 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 - 3*x1*x2^3*x3 + 2*x1*x2^2*x3 + x1*x2*x3^3 - 4*x1*x2*x3^2 + x1*x2*x3 - x1*x3^4 + x1*x3^3 + 2*x2^2*x3^2 - 2*x2*x3^3 + 2*x2*x3^2 + x3^4 - x3^3, x1^2*x2 - x1^2 - x1*x2 + x3 + 1, x2 - x3, x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 - x1*x3 + x2*x3 - x3^2 + x3, x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 - x1^2*x2*x3 + x1^2*x2 - x1^2*x3 + 2*x1*x2*x3 + 2*x1*x3 - x3^2 - x3, x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 - x1*x2^2 + x1*x2*x3 + x1*x2 + x1*x3^2 + x2*x3 - 2*x3^2, x1^3*x2^2 - x1^3*x2 - 2*x1^2*x2*x3 - x1*x2^2 + 3*x1*x2*x3 + x1*x2 + x2*x3 - 2*x3^2, x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 + x1^2*x2 - x1^2*x3 + x1*x2*x3 - x1*x3^2 + 2*x1*x3 - x3, x1*x3 - x1 + 1, x1^3*x2^2 - x1^3*x2 - x1^2*x2*x3 - x1*x2^2 + 2*x1*x2*x3 + x1*x2 - x1*x3^2 + x2*x3 - x3^2, x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 + x1^2*x3 - x1*x2^2 + x1*x2*x3 + x1*x2 - 2*x1*x3 + x2*x3 - x3^2 + x3, x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 - x1*x2^2 + x1*x2*x3 + x1*x2 + x2*x3 - x3^2, x1^3*x2^2 - x1^3*x2 - x1^2*x2*x3 - x1*x2^2 + 2*x1*x2*x3 + x1*x2 - x1*x3 + x2*x3 - x3^2 + x3, x1^2*x2^2 - x1^2*x2 - x1*x2^2 + x1*x2 + x1*x3^2 - x1*x3 + x2*x3 - x3^2 + x3, x1^3*x2^2 - x1^3*x2 - x1^2*x2^2 + x1^2*x2 + x1*x2*x3 - x1*x3 + x3, x1^2*x2 - x1^2 - x1*x2 + x1 + x3, x1^2*x2 - x1*x2 + x3, x1^4*x2^3 - x1^4*x2^2 - 2*x1^3*x2^3 - x1^3*x2^2*x3 + 2*x1^3*x2^2 + x1^2*x2^3 + 5*x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3^2 - 2*x1^2*x2*x3 - 4*x1*x2^2*x3 + 2*x1*x2*x3 + 3*x2*x3^2 - 2*x3^3, x1^4*x2^4 - 2*x1^4*x2^3 + x1^4*x2^2 - 2*x1^3*x2^4 + 4*x1^3*x2^3 - x1^3*x2^2*x3^2 - 2*x1^3*x2^2*x3 - 2*x1^3*x2^2 + x1^3*x2*x3^2 + 2*x1^3*x2*x3 + x1^2*x2^4 + 2*x1^2*x2^3*x3 - 2*x1^2*x2^3 + x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 + x1^2*x2^2 - 4*x1^2*x2*x3 - 2*x1*x2^3*x3 - x1*x2*x3^3 - 4*x1*x2*x3^2 + 2*x1*x2*x3 + 2*x1*x3^3 + x2^2*x3^2 + 3*x2*x3^2 - 2*x3^3, x1^2*x2^2 - x1^2*x2*x3 - x1^2*x2 - x1*x2^2 + x1*x2*x3 + x1*x2 + x2*x3 - 2*x3^2, x1^4*x2^3 - 2*x1^4*x2^2 + x1^4*x2 - 2*x1^3*x2^3 + 4*x1^3*x2^2 - x1^3*x2*x3^2 - 2*x1^3*x2 + x1^2*x2^3 + 2*x1^2*x2^2*x3 - 2*x1^2*x2^2 + x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x2 + x1^2*x3^2 - 2*x1*x2^2*x3 + 2*x1*x2*x3 - x1*x3^3 - x1*x3^2 + x2*x3^2, x1^4*x2^4 - 2*x1^4*x2^3 + x1^4*x2^2 - 2*x1^3*x2^4 + 4*x1^3*x2^3 - 2*x1^3*x2^2*x3 - 2*x1^3*x2^2 + 2*x1^3*x2*x3 + x1^2*x2^4 + 2*x1^2*x2^3*x3 - 2*x1^2*x2^3 - 2*x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 + x1^2*x2^2 + 3*x1^2*x2*x3^2 - 4*x1^2*x2*x3 - 2*x1*x2^3*x3 + 2*x1*x2^2*x3^2 - 6*x1*x2*x3^2 + 2*x1*x2*x3 + 2*x1*x3^3 + x2^2*x3^2 - 2*x2*x3^3 + 3*x2*x3^2 + x3^4 - 2*x3^3, x1^3*x2^2 - x1^3*x2 + x1^2*x2^2*x3 - 2*x1^2*x2^2 - 2*x1^2*x2*x3 + 2*x1^2*x2 - x1*x2^2*x3 + x1*x2^2 + 4*x1*x2*x3 - x1*x2 - x1*x3^2 + x2*x3^2 - 2*x2*x3 - x3^3 + x3^2]