The realization space is
  [0   1   1   0   0   1    1      x1^2*x2 - 2*x1*x2 - x1*x3 + x2*x3 + x3                x1^2*x2 - 2*x1*x2 - x1*x3 + x2*x3 + x3    1          x1 - x2 - 1]
  [1   0   1   1   0   1   x1               x1^3*x2 - 3*x1^2*x2 + 2*x1*x2   x1^3*x3 - x1^2*x2*x3 - 2*x1^2*x3 + x1*x2*x3 + x1*x3   x1    x1^2*x2 - 2*x1*x2]
  [1   0   1   0   1   0    1   x1^2*x3 - x1*x2*x3 - 2*x1*x3 + x2*x3 + x3             x1^2*x3 - x1*x2*x3 - 2*x1*x3 + x2*x3 + x3   x2   x1*x3 - x2*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[x1^2*x2 - x1^2 - x1*x3 + x1 - x2^2 + x2*x3 - x2 + x3, x1*x2^2 - x1*x3 - 2*x2^2 + x2*x3 + x3, x1, x1 - 1, x1*x2^2 - x1*x2 - x1*x3 + x1 - x2^2 + x2*x3 - x2 + x3, x2 - x3, x2 - 1, x1*x2^2 - x1*x3 - x2^2 + x2*x3 - x2 + x3, x1^2*x2 - x1^2 - x1*x2 - x1*x3 + 2*x1 + x2*x3 - x2 + x3 - 1, x3, x1 - 2, x2, x1*x2 - x2 - 1, x1*x2 - x1*x3 + x2*x3 - 2*x2 + x3, x3 - 1, x1*x2 - x1*x3 + x2*x3 - x2 + x3 - 1, x1^2*x2 - 2*x1*x2 - x1*x3 - x1 + x2*x3 + x2 + x3 + 1, x1^2*x2 - 2*x1*x2 - x1*x3 + x2*x3 + x3, x1^4*x2^2 - x1^4*x2 - 2*x1^3*x2^2 - x1^3*x2*x3 + 3*x1^3*x2 - x1^2*x2^3 + x1^2*x2^2*x3 - x1^2*x2^2 + 2*x1^2*x2*x3 - 2*x1^2*x2 + x1^2*x3 + 2*x1*x2^3 + 2*x1*x2^2 - 2*x1*x2*x3 - 2*x1*x3 - x2^3*x3 - x2^2*x3 + x2*x3 + x3, x1^3*x2^3 - x1^3*x3 - 4*x1^2*x2^3 - x1^2*x2^2*x3 + 2*x1^2*x2*x3 + 3*x1^2*x3 + x1*x2^3*x3 + 4*x1*x2^3 + 2*x1*x2^2*x3 - 4*x1*x2*x3 - 3*x1*x3 - 2*x2^3*x3 - x2^2*x3 + 2*x2*x3 + x3, x1^3*x2^3 - x1^3*x2^2 + x1^3*x2 - x1^3*x3 - 3*x1^2*x2^3 - x1^2*x2^2*x3 + x1^2*x2^2 + 3*x1^2*x2*x3 - 2*x1^2*x2 + 2*x1^2*x3 + x1*x2^3*x3 + 2*x1*x2^3 + 2*x1*x2^2 - 3*x1*x2*x3 - 2*x1*x3 - x2^3*x3 - x2^2*x3 + x2*x3 + x3, x1^2*x2^2 - x1^2*x3 - 2*x1*x2^2 + 2*x1*x3 + x2^2*x3 - x3, x1^3*x2^3 - x1^3*x3 - 3*x1^2*x2^3 - x1^2*x2^2*x3 - x1^2*x2^2 + 2*x1^2*x2*x3 + 3*x1^2*x3 + x1*x2^3*x3 + 2*x1*x2^3 + x1*x2^2*x3 + 2*x1*x2^2 - 3*x1*x2*x3 - 3*x1*x3 - x2^3*x3 - x2^2*x3 + x2*x3 + x3, x1^3*x2^2 - x1^3*x3 - 4*x1^2*x2^2 + x1^2*x2*x3 + x1^2*x2 + 2*x1^2*x3 + 4*x1*x2^2 - 2*x1*x2 - 2*x1*x3 - x2^2*x3 + x3, x1 - x2 - 1, x1 - x2, x1^2*x2^2 - x1*x2^2 - x1*x2*x3 - x1*x2 - x1*x3 + x2^2*x3 + 2*x2*x3 + x3, x1^4*x2^2 - 4*x1^3*x2^2 - x1^3*x2*x3 - x1^3*x3 + x1^2*x2^2*x3 + 4*x1^2*x2^2 + 5*x1^2*x2*x3 + 3*x1^2*x3 - 3*x1*x2^2*x3 - 6*x1*x2*x3 - 3*x1*x3 + x2^2*x3 + 2*x2*x3 + x3, x1^3*x2 - 4*x1^2*x2 + 4*x1*x2 + x1*x3 - x2*x3 - x3]