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