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