The realization space is
  [0   1   1   0   0   1    x1 - x2 + 1             x1 - x2 + 1                           -x1*x2 + 3*x1 + x2^2 - 1    -x1*x2 + 3*x1 + x2^2 - 1    1]
  [1   1   0   1   0   0   -x1 - x2 + 1   x1*x2 - x1 - x2^2 + 1   -x1*x2^2 + 6*x1*x2 - 5*x1 + x2^3 - x2^2 - x2 + 1   3*x1*x2 - 5*x1 - x2^2 + 1   x1]
  [1   1   0   0   1   1              0       x1*x2 - x2^2 + x2                                  x1*x2 + x2^2 - x2           x1*x2 + x2^2 - x2   x2]
in the multivariate polynomial ring in 2 variables over ZZ
within the vanishing set of the ideal
Ideal (x1^2 - 2*x1*x2 + 4*x1 + x2^2 - 1)
avoiding the zero loci of the polynomials
RingElem[2*x1^3 - 3*x1^2*x2 + 4*x1^2 + 2*x1*x2 - 2*x1 + x2^3 - 2*x2^2 + x2, x1^3*x2 - 2*x1^3 - 2*x1^2*x2^2 + 7*x1^2*x2 - 4*x1^2 + x1*x2^3 - 3*x1*x2 + 2*x1 - x2^3 + 2*x2^2 - x2, x2, x1 + x2 - 1, x1^2 - x1*x2 + 2*x1 + x2 - 1, x1^2*x2 - 2*x1^2 - 2*x1*x2^2 + 6*x1*x2 - 4*x1 + x2^3 - 3*x2 + 2, x1, x1^2 - 2*x1*x2 + 2*x1 + x2^2 - 1, 2*x1^3 - 5*x1^2*x2 + 4*x1^2 + 2*x1*x2^2 - 2*x1 + x2^3 - 2*x2^2 + x2, x1^2 - 3*x1*x2 + 2*x1 + x2 - 1, x1*x2 - 2*x1 - x2^2 + x2, x1^2 - 4*x1*x2 + 4*x1 + 3*x2^2 - 2*x2 - 1, x1^2*x2 - 2*x1^2 - 4*x1*x2^2 + 8*x1*x2 - 4*x1 + 3*x2^3 - 4*x2^2 - x2 + 2, x2 - 1, x1^2 - 2*x1*x2 + 2*x1 - x2^2 + 2*x2 - 1, x1 - x2 + 1, x1 - 1, x1 - x2, x1^2*x2 - 2*x1*x2^2 + 6*x1*x2 - 4*x1 + x2^3 - 2*x2^2 + x2, x1^2*x2 + x1^2 - 2*x1*x2^2 + 4*x1*x2 - 2*x1 + x2^3 - x2^2 - x2 + 1, 2, x1^2*x2 + 2*x1^2 - 2*x1*x2^2 + 4*x1*x2 - 2*x1 + x2^3 - 2*x2^2 + x2, x1 - x2 - 1, x1^2*x2 - 4*x1*x2 + 4*x1 - x2^3 + 2*x2^2 - x2]