The realization space is
  [1                         x3^2 - x3   1   0                                                      x3^2 - x3                                                                   x3^2 - x3                                                                   x3^2 - x3                         x3^2 - x3   0                                                      x3^2 - x3    0]
  [1   -x1*x2 - x1*x3 + x2 + x3^2 - x3   0   1   -x1*x2^2 - 2*x1*x2*x3 - x1*x3^2 + x2^2 + x2*x3^2 + x3^2 - x3   -x1*x2^2 - 2*x1*x2*x3 - x1*x3^2 + x2^2 + x2*x3^2 - x2*x3 + x2 + x3^2 - x3   -x1*x2^2 - 2*x1*x2*x3 - x1*x3^2 + x2^2 + x2*x3^2 - x2*x3 + x2 + x3^2 - x3   -x1*x2 - x1*x3 + x2 + x3^2 - x3   0   -x1*x2^2 - 2*x1*x2*x3 - x1*x3^2 + x2^2 + x2*x3^2 + x3^2 - x3    1]
  [1   -x1*x2 - x1*x3 + x2 + x3^2 - x3   0   0                                                      x3^2 - x3                                             -x1*x2 - x1*x3 + x2 + x3^2 - x3                                                                           0                   x1*x3^2 - x1*x3   1                                                x2*x3^2 - x2*x3   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 1 generator
avoiding the zero loci of the polynomials
RingElem[x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2 + x2*x3 - x2 - x3 + 1, x1^2*x2^2*x3 - x1^2*x2^2 + 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^3 - x1^2*x3^2 - x1*x2^3*x3 - 2*x1*x2^2*x3^2 - x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3^3 + x1*x2*x3^2 - x1*x3^3 + 2*x1*x3^2 - x1*x3 + x2^3*x3 + x2^2*x3^3 - x2^2*x3^2 + x2^2*x3 + x2*x3^3 - x2*x3^2, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2 + x2*x3 - x2 - x3^2 + x3, x3, x2 - 1, x1, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2 - x3^2 + x3, x1 - x2, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^2 - x1*x2*x3^2 + x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 + x2*x3^2 - 2*x2*x3 + x2, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3 - x3^2 + x3, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^3 - 2*x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3^2 + x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 + x2^3 + x2^2*x3^2 + x2*x3^2 - 2*x2*x3 + x2, x1^2*x2^2*x3 + x1^2*x2^2 + 2*x1^2*x2*x3^2 + 2*x1^2*x2*x3 + x1^2*x3^3 + x1^2*x3^2 - x1*x2^3*x3 - 2*x1*x2^2*x3^2 - x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3^3 - x1*x2*x3 - x1*x3^3 + x1*x3 + x2^3*x3 + x2^2*x3^3 - x2^2*x3^2 + x2^2*x3 + x2*x3^3 - x2*x3^2, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^3*x3 - 2*x1*x2^2*x3^2 - x1*x2^2 - x1*x2*x3^3 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2^3*x3 + x2^2*x3^3 - x2^2*x3^2 + x2^2*x3 + x2*x3^2 - x2*x3, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^3 - 2*x1*x2^2*x3 - x1*x2^2 - 2*x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2^3 + x2^2*x3^2 - x2^2*x3 + x2^2 + x2*x3^2 - x2*x3, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^3*x3 - 2*x1*x2^2*x3^2 - x1*x2^2 - x1*x2*x3^3 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2^3*x3 + x2^2*x3^3 - x2^2*x3^2 + x2^2*x3 + x2*x3^3 - x2*x3^2, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 - x2^2*x3^2 + x2^2*x3 + x2*x3^2 - x2*x3, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2, x1^2*x2^2*x3 - x1^2*x2^2 + 2*x1^2*x2*x3^2 - 2*x1^2*x2*x3 + x1^2*x3^3 - x1^2*x3^2 - x1*x2^3*x3 - 2*x1*x2^2*x3^2 - x1*x2^2*x3 + x1*x2^2 - 2*x1*x2*x3^3 + x1*x2*x3^2 - x1*x3^3 + 2*x1*x3^2 - x1*x3 + x2^3*x3 + x2^2*x3^3 + x2*x3^3 - 2*x2*x3^2 + x2*x3, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2 - x3^2 + 2*x3 - 1, x3 - 1, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2*x3^2 - x2*x3, x2, x1^3*x2^4 + 4*x1^3*x2^3*x3 + 6*x1^3*x2^2*x3^2 + 4*x1^3*x2*x3^3 + x1^3*x3^4 - 2*x1^2*x2^4 - 2*x1^2*x2^3*x3^2 - 4*x1^2*x2^3*x3 - 4*x1^2*x2^2*x3^3 - 4*x1^2*x2^2*x3^2 + 2*x1^2*x2^2*x3 - 2*x1^2*x2*x3^4 - 4*x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 - 2*x1^2*x3^4 + 2*x1^2*x3^3 + x1*x2^4*x3^2 - x1*x2^4*x3 + x1*x2^4 + 2*x1*x2^3*x3^3 + 2*x1*x2^2*x3^4 - x1*x2^2*x3^3 + 2*x1*x2^2*x3^2 - 2*x1*x2^2*x3 + 2*x1*x2*x3^4 - 2*x1*x2*x3^3 + x1*x3^4 - 2*x1*x3^3 + x1*x3^2 - x2^4*x3^2 + x2^4*x3 - x2^3*x3^4 + 2*x2^3*x3^3 - 2*x2^3*x3^2 + x2^3*x3 - x2^2*x3^4 + 2*x2^2*x3^3 - x2^2*x3^2, x1^3*x2^4 + 4*x1^3*x2^3*x3 + 6*x1^3*x2^2*x3^2 + 4*x1^3*x2*x3^3 + x1^3*x3^4 - 2*x1^2*x2^4 - 2*x1^2*x2^3*x3^2 - 4*x1^2*x2^3*x3 - 4*x1^2*x2^2*x3^3 - 3*x1^2*x2^2*x3^2 + x1^2*x2^2*x3 - 2*x1^2*x2*x3^4 - 2*x1^2*x2*x3^3 + 2*x1^2*x2*x3^2 - x1^2*x3^4 + x1^2*x3^3 + x1*x2^4*x3^2 - x1*x2^4*x3 + x1*x2^4 + 2*x1*x2^3*x3^3 + 2*x1*x2^2*x3^4 - x1*x2^2*x3^3 + x1*x2^2*x3^2 - x1*x2^2*x3 + x1*x2*x3^4 - x1*x2*x3^3 - x2^4*x3^2 + x2^4*x3 - x2^3*x3^4 + 2*x2^3*x3^3 - 2*x2^3*x3^2 + x2^3*x3 - x2^2*x3^3 + 2*x2^2*x3^2 - x2^2*x3, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^2 - x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2^2*x3^2 - x2^2*x3, x1*x2^2 + 2*x1*x2*x3 + x1*x3^2 - x2^2 - x2*x3^2 + x2*x3 - x2, x1^2*x2^2 + 2*x1^2*x2*x3 + x1^2*x3^2 - x1*x2^3 - x1*x2^2*x3 - 2*x1*x2^2 - 2*x1*x2*x3^2 - x1*x3^2 + x1*x3 + x2^3 + x2^2*x3^2 - x2^2*x3 + x2^2 + x2*x3^2 - x2*x3, x1^4*x2^4 + 4*x1^4*x2^3*x3 + 6*x1^4*x2^2*x3^2 + 4*x1^4*x2*x3^3 + x1^4*x3^4 - 2*x1^3*x2^4 - 2*x1^3*x2^3*x3^2 - 4*x1^3*x2^3*x3 - 4*x1^3*x2^2*x3^3 - 4*x1^3*x2^2*x3^2 + 2*x1^3*x2^2*x3 - 2*x1^3*x2*x3^4 - 4*x1^3*x2*x3^3 + 4*x1^3*x2*x3^2 - 2*x1^3*x3^4 + 2*x1^3*x3^3 + x1^2*x2^4*x3^2 - x1^2*x2^4*x3 + x1^2*x2^4 + 2*x1^2*x2^3*x3^3 + 2*x1^2*x2^3*x3^2 - 2*x1^2*x2^3*x3 + 2*x1^2*x2^2*x3^4 + 3*x1^2*x2^2*x3^3 - 2*x1^2*x2^2*x3^2 - 2*x1^2*x2^2*x3 + 4*x1^2*x2*x3^4 - 4*x1^2*x2*x3^3 + x1^2*x3^4 - 2*x1^2*x3^3 + x1^2*x3^2 - 2*x1*x2^4*x3^2 + 2*x1*x2^4*x3 - x1*x2^3*x3^4 - 2*x1*x2^3*x3^2 + 3*x1*x2^3*x3 - 3*x1*x2^2*x3^4 + 3*x1*x2^2*x3^3 - 2*x1*x2*x3^4 + 4*x1*x2*x3^3 - 2*x1*x2*x3^2 + x2^4*x3^2 - x2^4*x3 + x2^3*x3^4 - 2*x2^3*x3^3 + 2*x2^3*x3^2 - x2^3*x3 + x2^2*x3^4 - 2*x2^2*x3^3 + x2^2*x3^2, x1^4*x2^4 + 4*x1^4*x2^3*x3 + 6*x1^4*x2^2*x3^2 + 4*x1^4*x2*x3^3 + x1^4*x3^4 - x1^3*x2^5 - 4*x1^3*x2^4*x3 - 2*x1^3*x2^4 - 8*x1^3*x2^3*x3^2 - 4*x1^3*x2^3*x3 - 8*x1^3*x2^2*x3^3 - 4*x1^3*x2^2*x3^2 + 2*x1^3*x2^2*x3 - 3*x1^3*x2*x3^4 - 4*x1^3*x2*x3^3 + 4*x1^3*x2*x3^2 - 2*x1^3*x3^4 + 2*x1^3*x3^3 + 2*x1^2*x2^5 + 2*x1^2*x2^4*x3^2 + 4*x1^2*x2^4*x3 + x1^2*x2^4 + 4*x1^2*x2^3*x3^3 + 7*x1^2*x2^3*x3^2 - 3*x1^2*x2^3*x3 + 3*x1^2*x2^2*x3^4 + 6*x1^2*x2^2*x3^3 - 4*x1^2*x2^2*x3^2 - 2*x1^2*x2^2*x3 + 5*x1^2*x2*x3^4 - 5*x1^2*x2*x3^3 + x1^2*x3^4 - 2*x1^2*x3^3 + x1^2*x3^2 - x1*x2^5 - 3*x1*x2^4*x3^2 + x1*x2^4*x3 - x1*x2^3*x3^4 - x1*x2^3*x3^3 - 3*x1*x2^3*x3^2 + 4*x1*x2^3*x3 - 4*x1*x2^2*x3^4 + 4*x1*x2^2*x3^3 - 2*x1*x2*x3^4 + 4*x1*x2*x3^3 - 2*x1*x2*x3^2 + x2^4*x3^2 - x2^4*x3 + x2^3*x3^4 - 2*x2^3*x3^3 + 2*x2^3*x3^2 - x2^3*x3 + x2^2*x3^4 - 2*x2^2*x3^3 + x2^2*x3^2, x1 - 1, x1^2*x2^3 + x1^2*x2^2*x3 - x1^2*x2*x3^2 - x1^2*x3^3 + x1*x2^3*x3 - x1*x2^3 + x1*x2^2*x3^2 + x1*x2^2*x3 + 2*x1*x2*x3^3 - x1*x2*x3^2 + x1*x2*x3 + x1*x3^3 - x1*x3^2 - x2^3*x3 - x2^2*x3^3 + x2^2*x3^2 - x2^2*x3 - x2*x3^3 + x2*x3^2]