The realization space is
  [1   1   0   0   1   1                               0                                                           x2 - 1                                                                    x2 - 1    1    1]
  [0   1   1   0   0   1                          x2 - 1   -x1*x3^2 + 2*x1*x3 - x1 + x2^2 + x2*x3^2 - 4*x2*x3 + x2 + x3^2   -x1*x3^2 + 3*x1*x3 - 2*x1 + x2^2 + x2*x3^2 - 4*x2*x3 + 2*x2 + x3^2 - x3   x1   x2]
  [0   0   0   1   1   1   x1*x3 - x1 - x2*x3 + 2*x2 - 1                                             x1*x3 - x1 + x2 - x3                                         x1*x3 - x1 + x2 + x3^2 - 3*x3 + 1   x3   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 3 generators
avoiding the zero loci of the polynomials
RingElem[x1 - x2, x3, x3 - 1, x1 - x2 + x3 - 1, x2 - x3, x1*x2 - x1*x3 - x2^2 + 3*x2*x3 - x2 - x3^2, x2, x1*x2^2 - x1*x2*x3 - x2^3 + 2*x2^2*x3 - x2^2 + x2*x3 - x3^2, x1*x2^2 + x1*x2*x3^2 - x1*x2*x3 - x1*x3^2 - x2^3 - x2^2*x3^2 + 3*x2^2*x3 - x2^2, x1*x2 + x1*x3^2 - x1*x3 - x2^2 - x2*x3^2 + 3*x2*x3 - x2 - x3^2, x1*x2^2 + x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^2 + x1*x3 - x2^3 - x2^2*x3^2 + 4*x2^2*x3 - x2^2 - x2*x3, x1^2*x2 - x1^2*x3 - 2*x1*x2^2 + 3*x1*x2*x3 - 2*x1*x2 - x1*x3^2 + 2*x1*x3 + x2^3 - 2*x2^2*x3 + 2*x2^2 + x2*x3^2 - 4*x2*x3 + x2 + x3^2, x1*x2 - x1*x3 - x2^2 + 2*x2*x3 - x2 + x3, x1*x2^2 - x1*x2*x3 - x2^3 + 2*x2^2*x3 - x2^2 - x3^2 + x3, x1*x2 - x1*x3 - x2^2 + 2*x2*x3 - x2, x2 - 1, x2 + x3 - 1, x1*x3 - x1 + x2 + x3^2 - 3*x3 + 1, x1^2*x3 - x1^2 - x1*x2*x3 + 2*x1*x2 + x1*x3^2 - 4*x1*x3 + 2*x1 - x2^2 - x2*x3^2 + 4*x2*x3 - 2*x2 - x3^2 + 2*x3 - 1, x1*x2 - x1*x3 - x2^2 - x2*x3^2 + 3*x2*x3 - x2, x1^2*x3 - x1*x2*x3 + x1*x2 + x1*x3^2 - 2*x1*x3 - x2^2 - x2*x3^2 + 3*x2*x3 - x2, x1^2*x3 - x1*x2*x3 + x1*x2 + x1*x3^2 - 3*x1*x3 - x2^2 - x2*x3^2 + 4*x2*x3 - x2 - x3^2 + x3, x1*x2^2 - 2*x1*x2*x3 + x1*x3^2 - x2^3 - x2^2*x3^2 + 4*x2^2*x3 - x2^2 - 2*x2*x3^2 + x2*x3 + x3^3 - x3^2, x1, x1 - 1, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 + x2*x3^2 - x3^2, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - 2*x1*x2 - x1*x3^2 + 2*x1*x3 + x2^2 + x2*x3^2 - 2*x2*x3 + x2 - x3, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - x1*x2 - x1*x3^2 + x1*x3 + x2*x3^2 - x2*x3 - x3^2 + x3, x1 - x3, x1 + x3 - 1, x1^2*x2^2*x3 - x1^2*x2^2 + x1^2*x2*x3^3 - 2*x1^2*x2*x3^2 + x1^2*x2*x3 - x1^2*x3^3 + x1^2*x3^2 - x1*x2^3*x3 + 2*x1*x2^3 - x1*x2^2*x3^3 + 6*x1*x2^2*x3^2 - 9*x1*x2^2*x3 + 2*x1*x2^2 + x1*x2*x3^4 - 5*x1*x2*x3^3 + 5*x1*x2*x3^2 - x1*x3^4 + 4*x1*x3^3 - 3*x1*x3^2 - x2^4 - 2*x2^3*x3^2 + 7*x2^3*x3 - 2*x2^3 - x2^2*x3^4 + 7*x2^2*x3^3 - 15*x2^2*x3^2 + 6*x2^2*x3 - x2^2 - x2*x3^4 + 2*x2*x3^3 + 3*x2*x3^2 - x2*x3 + x3^4 - 3*x3^3 + x3^2, x1*x2 + x1*x3^2 - 2*x1*x3 - x2^2 - x2*x3^2 + 4*x2*x3 - x2 + x3^3 - 3*x3^2 + x3, x1^2*x2^2*x3 - x1^2*x2^2 + x1^2*x2*x3^3 - 2*x1^2*x2*x3^2 + x1^2*x2*x3 - x1^2*x3^3 + x1^2*x3^2 - x1*x2^3*x3 + 2*x1*x2^3 - x1*x2^2*x3^3 + 6*x1*x2^2*x3^2 - 10*x1*x2^2*x3 + 2*x1*x2^2 + x1*x2*x3^4 - 5*x1*x2*x3^3 + 6*x1*x2*x3^2 + x1*x2*x3 - x1*x3^4 + 4*x1*x3^3 - 4*x1*x3^2 - x2^4 - 2*x2^3*x3^2 + 8*x2^3*x3 - 2*x2^3 - x2^2*x3^4 + 8*x2^2*x3^3 - 18*x2^2*x3^2 + 6*x2^2*x3 - x2^2 - x2*x3^4 + x2*x3^3 + 6*x2*x3^2 - 2*x2*x3 + x3^4 - 3*x3^3 + x3^2, x1^2*x2^2 - 2*x1^2*x2*x3 + x1^2*x3^2 - 2*x1*x2^3 - x1*x2^2*x3^2 + 6*x1*x2^2*x3 - 2*x1*x2^2 - 4*x1*x2*x3^2 + 4*x1*x2*x3 + 2*x1*x3^3 - 3*x1*x3^2 + x2^4 + x2^3*x3^2 - 4*x2^3*x3 + 2*x2^3 - x2^2*x3^3 + 4*x2^2*x3^2 - 6*x2^2*x3 + x2^2 - 2*x2*x3^3 + 7*x2*x3^2 - 2*x2*x3 + x3^4 - 3*x3^3 + x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3^2 + x1^2*x2*x3 - x1*x2^3*x3 + 2*x1*x2^3 + 3*x1*x2^2*x3^2 - 7*x1*x2^2*x3 + 2*x1*x2^2 - x1*x2*x3^3 + 4*x1*x2*x3^2 - 2*x1*x2*x3 - x1*x3^3 + x1*x3^2 - x2^4 - x2^3*x3^2 + 5*x2^3*x3 - 2*x2^3 + 2*x2^2*x3^3 - 8*x2^2*x3^2 + 6*x2^2*x3 - x2^2 + x2*x3^3 - 2*x2*x3^2 + x2*x3 - x3^4 + 3*x3^3 - 2*x3^2, x1^2*x2^2*x3 - x1^2*x2^2 - x1^2*x2*x3^2 + x1^2*x2*x3 - x1*x2^3*x3 + 2*x1*x2^3 + 3*x1*x2^2*x3^2 - 8*x1*x2^2*x3 + 2*x1*x2^2 - x1*x2*x3^3 + 5*x1*x2*x3^2 - x1*x2*x3 - x1*x3^3 - x2^4 - x2^3*x3^2 + 6*x2^3*x3 - 2*x2^3 + 2*x2^2*x3^3 - 10*x2^2*x3^2 + 6*x2^2*x3 - x2^2 + 2*x2*x3^3 - x2*x3^2 - x3^4 + 2*x3^3 - x3^2, x1*x2*x3 - x1*x2 + x2^2 + x2*x3^2 - 4*x2*x3 + x2 + x3, x1 + x3 - 2, x1*x2*x3 - x1*x2 + x1*x3^2 - x1*x3 + x2^2 + x2*x3^2 - 3*x2*x3 + x2 + x3^3 - 3*x3^2 + 2*x3, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - 3*x1*x2 - x1*x3^2 + 3*x1*x3 + 2*x2^2 + x2*x3^2 - 5*x2*x3 + 2*x2 + x3^2 - x3, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - 2*x1*x2 - x1*x3^2 + 2*x1*x3 + x2^2 + x2*x3^2 - 4*x2*x3 + x2 + x3, x1^2*x2 - x1^2*x3 - x1*x2^2 + 2*x1*x2*x3 - 3*x1*x2 - x1*x3^2 + 3*x1*x3 + 2*x2^2 + x2*x3^2 - 6*x2*x3 + 2*x2 + x3^2, x1*x2*x3 + x1*x2 - 2*x1*x3 - x2^2 - x2*x3^2 + 3*x2*x3 - x2, x1*x2 - x1*x3 - x2^2 - x2*x3^2 + 4*x2*x3 - x2 - x3, x1*x2 - x1*x3 - x2^2 + x2*x3 - x2 - x3^2 + 2*x3]