The realization space is
  [0   1   1   0   0   1                                                                                     1                                                                                        1    1                                                                                         x3    1]
  [1   0   1   1   0   1   -2*x1*x3^2 - x1*x3 + 3*x1 + x2^2*x3 - x2^2 + 3*x2*x3^2 - 4*x2*x3 + 2*x2 + x3^2 - x3                                                                                       x1   x1                                                                                      x2*x3   x2]
  [1   0   1   0   1   0                                                                                     1   2*x1*x3^2 + x1*x3 - 2*x1 - x2^2*x3 + x2^2 - 3*x2*x3^2 + 4*x2*x3 - 2*x2 - x3^2 + x3 + 1   x3   2*x1*x3^2 + 2*x1*x3 - 2*x1 - x2^2*x3 + x2^2 - 3*x2*x3^2 + 3*x2*x3 - 2*x2 - x3^2 + x3 + 1   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 5 generators
avoiding the zero loci of the polynomials
RingElem[x1*x2 - x2^2 - 2*x2*x3 + 2*x2 + x3^2 - 1, x2, x2 - 1, x1*x2 + x1*x3 + x1 - x2^2 - 3*x2*x3 + x2, x1 - x2, x3, x3 - 1, x1*x2 + x1*x3 - x2^2 - 3*x2*x3 + 2*x2 + x3 - 1, x1*x2 + x1*x3 + x1 - x2^2 - 3*x2*x3 + x2 + x3 - 1, x1*x2^2 + x1*x2*x3 + x1*x3 - x2^3 - 3*x2^2*x3 + 2*x2^2 - x2, x1*x2^2 + x1*x2*x3 - x1*x2 - x1 - x2^3 - 3*x2^2*x3 + 3*x2^2 + 3*x2*x3 - 2*x2 - x3 + 1, x1*x2 + x1*x3 - x2^2 - 3*x2*x3 + 2*x2, x1*x2*x3 + x1*x3^2 + x1*x3 - x2^2*x3 - 3*x2*x3^2 + 2*x2*x3 - x2, x1*x2 + x1*x3 + x1 - x2^2 - 3*x2*x3 + 2*x2 - 1, x1*x2*x3 + x1*x3^2 + x1*x3 - x2^2*x3 - 3*x2*x3^2 + 2*x2*x3 - x2 - x3 + 1, x2 - x3 - 1, x2 - x3, x1*x2 - x1*x3 - x2^2 - x2*x3 + 2*x2 + x3^2 - 1, x1^2*x2 - x1*x2^2 - 2*x1*x2*x3 + 2*x1*x2 - x1 + x2*x3^2, x1^2*x2 - x1*x2^2 - 2*x1*x2*x3 + x1*x2 + x1*x3 - x1 + x2^2 + x2*x3^2 + x2*x3 - 2*x2 - x3^2 + 1, x1^2*x2 - x1*x2^2 - 2*x1*x2*x3 + x1*x2 - x1 + x2^2 + x2*x3^2 + 2*x2*x3 - 2*x2 - x3^2 + 1, x1^2*x2^2 + x1^2*x2*x3 - 2*x1*x2^3 - 6*x1*x2^2*x3 + 4*x1*x2^2 - x1*x2*x3^2 + 2*x1*x2*x3 - x1*x2 + x1*x3^3 + x1*x3^2 - 2*x1*x3 + x2^4 + 5*x2^3*x3 - 4*x2^3 + 5*x2^2*x3^2 - 10*x2^2*x3 + 5*x2^2 - 3*x2*x3^3 + x2*x3^2 + 4*x2*x3 - 2*x2, x1*x2*x3 - x1*x2 + x1*x3^2 + x1*x3 - x2^2*x3 + x2^2 - 3*x2*x3^2 + 3*x2*x3 - 2*x2 - x3 + 1, x1^2*x2 - x1*x2^2*x3 - x1*x2^2 - x1*x2*x3^2 - x1*x2*x3 + x1*x2 + x1*x3^2 + x1*x3 - x1 + x2^3*x3 + 3*x2^2*x3^2 - 3*x2^2*x3 + x2^2 - 3*x2*x3^2 + 4*x2*x3 - 2*x2 - x3 + 1, x1*x2*x3 - x1*x2 + x1*x3^2 - x2^2*x3 + x2^2 - 3*x2*x3^2 + 4*x2*x3 - 2*x2 - x3 + 1, x1^2*x2 - x1*x2^2*x3 - x1*x2^2 - x1*x2*x3^2 - x1*x2*x3 + x1*x2 + x1*x3^2 - x1 + x2^3*x3 + 3*x2^2*x3^2 - 3*x2^2*x3 + x2^2 - 3*x2*x3^2 + 5*x2*x3 - 2*x2 - x3 + 1, x1^2*x2^2 + x1^2*x2*x3 + x1^2*x2 - 2*x1*x2^3 - 6*x1*x2^2*x3 + 3*x1*x2^2 - 2*x1*x2*x3^2 + x1*x2 - x1*x3 - x1 + x2^4 + 5*x2^3*x3 - 4*x2^3 + 6*x2^2*x3^2 - 10*x2^2*x3 + 5*x2^2 + 4*x2*x3 - 2*x2, x1*x2 - x2^2 - 2*x2*x3 + 2*x2 + x3 - 1, x1*x2 - x2^2 - x2*x3 + 2*x2 - x3 - 1, x1*x2 - x2^2 - 2*x2*x3 + 2*x2 - 1, x1*x2 - x2^2 - x2*x3 + 2*x2 - 1, x1, x1 - 1, x1*x2*x3 + x1*x3^2 + x1*x3 - x1 - x2^2*x3 - 3*x2*x3^2 + 2*x2*x3 - x3 + 1, x1 - x3 - 1, x1 - x3, x1*x2 + x1*x3 + x1 - x2^2 - 3*x2*x3 + 2*x2 - 2, x1*x2 + x1*x3 - x2^2 - 3*x2*x3 + 2*x2 - 1, x1*x2 + x1*x3 + x1 - x2^2 - 3*x2*x3 + 2*x2]