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