The realization space is
  [1   1   0   0   1   1                                                       0                 x3                                           x2*x3^2 - x3^2                       x3^2    1]
  [1   0   1   0   1   0                                            x1^2 - x1*x3   -x1 + x2*x3 + x3                                      x2^2*x3^2 - x2*x3^2   -x1^2 + x1*x2*x3 + x1*x3   x2]
  [0   0   0   1   1   1   -x1^2 + x1*x2*x3 - 2*x1*x3^2 + x1*x3 - x2*x3^2 + x3^3               x3^2   -x1^2 + x1*x2*x3 - x1*x3^2 + x1*x3 + x2*x3^3 - x2*x3^2                    x1*x3^2   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 - x3, x1, x1 - x2*x3, x1*x3 - x1 + x2*x3 - x3^2, x1*x3 - x1 + x2*x3, x1 - x2*x3 + x3^2, x2 - 1, x2, x3, x3 - 1, x1^2 - x1*x2*x3 + 2*x1*x3^2 - x1*x3 + x2*x3^2 - x3^3, x1^2*x2 + x1^2*x3 - x1^2 - x1*x2^2*x3 + 2*x1*x2*x3^2 - 3*x1*x3^2 + x1*x3 + x2^2*x3^2 - x2*x3^3 - x2*x3^2 + x3^3, x1^2*x2 + x1^2*x3 - x1*x2^2*x3 + 2*x1*x2*x3^2 - x1*x2*x3 - x1*x3^2 + x2^2*x3^2 - x2*x3^3, x1^2*x2 + x1^2*x3 - 2*x1^2 - x1*x2^2*x3 + 2*x1*x2*x3^2 - 3*x1*x3^2 + 2*x1*x3 + x2^2*x3^2 - x2*x3^3 - x2*x3^2 + x3^3, x1^2*x2 + x1^2*x3 - x1^2 - x1*x2^2*x3 + 2*x1*x2*x3^2 - x1*x2*x3 - x1*x3^2 + x1*x3 + x2^2*x3^2 - x2*x3^3, x2 + x3 - 1, x2 - x3, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 - x1^2*x3 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + x1*x2*x3^2 + x1*x3^2 + x2*x3^4 - x2*x3^3, x1^2 - 2*x1*x2*x3 + x1*x3^2 - x1*x3 + x2^2*x3^2 - 2*x2*x3^3 + x2*x3^2, x1 + x2*x3^2 - x2*x3, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 - x1^2*x3 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + 2*x1*x2*x3^2 - x1*x3^2 - x2^2*x3^3 + x2*x3^3, x1^3*x2 - 2*x1^3 - 2*x1^2*x2^2*x3 + x1^2*x2*x3^2 + 2*x1^2*x2*x3 - 2*x1^2*x3^2 + 2*x1^2*x3 + x1*x2^3*x3^2 - 2*x1*x2^2*x3^3 + x1*x2^2*x3^2 + 2*x1*x2*x3^3 - 3*x1*x2*x3^2 + x1*x3^3 - x2^3*x3^3 + x2^2*x3^4 + x2^2*x3^3 - x2*x3^4, x1 - x2*x3 - x3, x1 - x2*x3 + x3^2 - x3, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 - 2*x1^2*x3 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x3^3 + x1*x3^2 - x2^2*x3^3 + x2*x3^4 - x2*x3^3 + x3^4, x1^4 - 2*x1^3*x2*x3 + x1^3*x3^2 - 2*x1^3*x3 + x1^2*x2^2*x3^2 - 2*x1^2*x2*x3^3 + 3*x1^2*x2*x3^2 - 2*x1^2*x3^3 + 2*x1^2*x3^2 - x1*x2^2*x3^3 + x1*x2*x3^4 - 2*x1*x2*x3^3 + 3*x1*x3^4 - x1*x3^3 + x2*x3^4 - x3^5, x1^4 - 2*x1^3*x2*x3 + x1^3*x3^2 - 2*x1^3*x3 + x1^2*x2^2*x3^2 - 2*x1^2*x2*x3^3 + 3*x1^2*x2*x3^2 - 2*x1^2*x3^3 + 3*x1^2*x3^2 - x1*x2^2*x3^3 + x1*x2*x3^4 - 2*x1*x2*x3^3 + 3*x1*x3^4 - 2*x1*x3^3 + x2*x3^4 - x3^5, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 - 2*x1^2*x3 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x3^3 + 2*x1*x3^2 - x2^2*x3^3 + x2*x3^4 - x2*x3^3 + x3^4 - x3^3, x1^2 - x1*x2*x3 - x1*x3^2 - x1*x3 + x3^2, x1 - 1, x1^2 - x1*x2*x3 - x1*x3 + x3^2, x1^2 - 2*x1*x2*x3 + x1*x3^2 + x2^2*x3^2 - 2*x2*x3^3 + x3^3, x1^2 - 2*x1*x2*x3 + x1*x3^2 + x2^2*x3^2 - 2*x2*x3^3 + x2*x3^2 + x3^3 - x3^2, x1^2 - 2*x1*x2*x3 + x1*x3^2 - x1*x3 + x2^2*x3^2 - 2*x2*x3^3 + 2*x2*x3^2 - x3^2, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + x1*x2*x3^2 + x1*x3^3 - x1*x3^2 - x2^2*x3^3 + x2*x3^4 + x2*x3^3 - x3^4, x1^4 - x1^3*x2*x3 + x1^3*x3^2 - 2*x1^3*x3 - x1^2*x2^2*x3^2 - x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 - x1^2*x3^3 + x1*x2^3*x3^3 - 2*x1*x2^2*x3^4 - x1*x2^2*x3^3 + 5*x1*x2*x3^4 - 2*x1*x2*x3^3 - 2*x1*x3^4 + x1*x3^3 - x2^3*x3^4 + x2^2*x3^5 + 2*x2^2*x3^4 - 2*x2*x3^5 - x2*x3^4 + x3^5, x1^4 - x1^3*x2*x3 + x1^3*x3^2 - 2*x1^3*x3 - x1^2*x2^2*x3^2 - x1^2*x2*x3^3 + 5*x1^2*x2*x3^2 - x1^2*x3^3 - x1^2*x3^2 + x1*x2^3*x3^3 - 2*x1*x2^2*x3^4 - x1*x2^2*x3^3 + 5*x1*x2*x3^4 - 3*x1*x2*x3^3 - 2*x1*x3^4 + 2*x1*x3^3 - x2^3*x3^4 + x2^2*x3^5 + 2*x2^2*x3^4 - 2*x2*x3^5 - x2*x3^4 + x3^5, x1^4 - x1^3*x2*x3 + x1^3*x3^2 - 2*x1^3*x3 - x1^2*x2^2*x3^2 - x1^2*x2*x3^3 + 4*x1^2*x2*x3^2 - x1^2*x3^3 + x1*x2^3*x3^3 - 2*x1*x2^2*x3^4 + 3*x1*x2*x3^4 - 3*x1*x2*x3^3 + x1*x3^3 - x2^3*x3^4 + x2^2*x3^5 + x2^2*x3^4 - x2*x3^5, x1^2 - x1*x2*x3 + x1*x3^2 - x1*x3 - x2^2*x3^2 - x2*x3^3 + 3*x2*x3^2 - x3^2, x1^2 - x1*x2*x3 + x1*x3^2 - x1*x3 - x2*x3^3 + 2*x2*x3^2 - x3^2, x1^2 - x1*x2*x3 + x1*x3^2 - x1*x3 + x2^2*x3^2 - x2*x3^3, x1 + x3^2 - x3, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + 2*x1*x2*x3^2 - x1*x3^2 - x2^2*x3^3 + x2*x3^4, x1^3 - 2*x1^2*x2*x3 + x1^2*x3^2 - x1^2*x3 + x1*x2^2*x3^2 - 2*x1*x2*x3^3 + 3*x1*x2*x3^2 - 2*x1*x3^3 - x2^2*x3^3 + x2*x3^4 - x2*x3^3 + x3^4, x1 - x2*x3 - x3^2, x1*x2 - 2*x1*x3 - x2*x3 + x3^2, 2*x1^2 - x1*x2*x3 + 2*x1*x3^2 - 2*x1*x3 + x2*x3^2 - x3^3]