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