The realization space is
  [1   1   0   1   0    0    1                                                                                                     x1^3 + 2*x1^2*x2 - x1^2 + x1*x2^2 - x1*x2                           x1    1                             x1 + x2 - 1]
  [0   1   1   0   0    1   x1                         2*x1^3*x2 - x1^3 + 3*x1^2*x2^2 - 4*x1^2*x2 + 2*x1^2 + x1*x2^3 - 4*x1*x2^2 + 4*x1*x2 - 2*x1 - x2^3 + 2*x2^2 - 2*x2 + 1   x1^2 + x1*x2 - x1 - x2 + 1   x1   x1^2*x2 + x1*x2^2 - x1*x2 - x2^2 + x2]
  [0   0   0   1   1   -1    1   x1^3*x2^2 - x1^3*x2 + x1^3 + x1^2*x2^3 - 2*x1^2*x2^2 + 4*x1^2*x2 - 2*x1^2 - x1*x2^3 + 4*x1*x2^2 - 5*x1*x2 + 2*x1 + x2^3 - 3*x2^2 + 3*x2 - 1                  x1 + x2 - 1   x2                       x1*x2 + 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 - x1^3 - x1^2*x2 + x1 + x2 - 1)
avoiding the zero loci of the polynomials
RingElem[x1 - 1, x2 - 1, x1 + x2, x2, x1^2 + x1*x2 - x1 - x2 + 1, x1, x1^2*x2 + x1*x2^2 - x1 - x2 + 1, x1^2 + x1*x2 - 1, x1*x2 + x2^2 - 1, x1 + x2 - 1, x1^2 - x1 - x2 + 1, x1^3 + x1^2*x2 - 2*x1^2 - 3*x1*x2 + 2*x1 - x2^2 + 2*x2 - 1, x1^3 + x1^2*x2 - x1^2 - 2*x1*x2 + 2*x1 - x2^2 + 2*x2 - 1, x1^3 + x1^2*x2 - x1^2 - x1*x2 + x1 + x2 - 1, x1^3*x2 - x1^3 + x1^2*x2^2 - 2*x1^2*x2 + 2*x1^2 - x1*x2^2 + 3*x1*x2 - 2*x1 + x2^2 - 2*x2 + 1, x1 + 1]