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