The realization space is
  [1   0   1   0   1    0                                                                                         x1*x2                                                  x2                                              x2^2 - x2                                                           x1*x2    1]
  [0   1   1   0   0    1                                                                                         x1*x2   x1^2*x2 - x1^2 + x1*x2^2 - x1*x2 + x1 - x2^2 + x2   -x1^2*x2 + x1^2 - x1*x2^2 + x1*x2 - x1 + 2*x2^2 - x2   x1^3*x2 - x1^3 + x1^2*x2^2 - x1^2*x2 + x1^2 - x1*x2^2 + x1*x2   x1]
  [0   0   0   1   1   -1   x1^3*x2 - x1^3 + 2*x1^2*x2^2 - 2*x1^2*x2 + x1^2 + x1*x2^3 - 2*x1*x2^2 + x1*x2 - x2^3 + x2^2                                                x2^2                                -x1*x2^2 + x1*x2 - x2^3   x1^2*x2^2 - x1^2*x2 + x1*x2^3 - x1*x2^2 + x1*x2 - x2^3 + x2^2   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 - 1, x2 - 1, x1 + x2, x1, x1*x2 - x1 + x2^2, x1^2*x2 - x1^2 + 2*x1*x2^2 - x1*x2 + x2^3 - x2^2, x1*x2 - x1 + 2*x2^2 - x2, x1*x2 - x1 + x2^2 + x2 - 1, x1^2*x2^2 - 3*x1^2*x2 + 2*x1^2 + 2*x1*x2^3 - 5*x1*x2^2 + 2*x1*x2 + x2^4 - 2*x2^3 + x2^2, x2, x1^3*x2 - x1^3 + 2*x1^2*x2^2 - 2*x1^2*x2 + x1^2 + x1*x2^3 - 2*x1*x2^2 + x1*x2 - x2^3 + x2^2, x1^2*x2 - x1^2 + 2*x1*x2^2 - x1*x2 + x2^3 - x2, x1 + x2 - 1, x1^2*x2 - x1^2 + x1*x2^2 - x1*x2 + x1 - x2^2 + x2, x1^3*x2^2 - 2*x1^3*x2 + x1^3 + 2*x1^2*x2^3 - 3*x1^2*x2^2 + 2*x1^2*x2 - x1^2 + x1*x2^4 - 2*x1*x2^3 + 2*x1*x2^2 - x2^4 + x2^3, x1^2*x2 - x1^2 + x1*x2^2 + x1 - x2^2, x1^2 + x1*x2 - x2, x1^4*x2^3 - 3*x1^4*x2^2 + 3*x1^4*x2 - x1^4 + 3*x1^3*x2^4 - 8*x1^3*x2^3 + 8*x1^3*x2^2 - 4*x1^3*x2 + x1^3 + 3*x1^2*x2^5 - 8*x1^2*x2^4 + 11*x1^2*x2^3 - 8*x1^2*x2^2 + 2*x1^2*x2 + x1*x2^6 - 4*x1*x2^5 + 9*x1*x2^4 - 7*x1*x2^3 + 2*x1*x2^2 - x2^6 + 3*x2^5 - 3*x2^4 + x2^3, x1^3*x2^2 - 2*x1^3*x2 + x1^3 + 2*x1^2*x2^3 - 4*x1^2*x2^2 + 3*x1^2*x2 - x1^2 + x1*x2^4 - 3*x1*x2^3 + 4*x1*x2^2 - x1*x2 - x2^4 + 2*x2^3 - x2^2, x1^2*x2 - x1^2 - x1*x2^3 + 3*x1*x2^2 - x1*x2 - x2^4 + 2*x2^3 - x2^2, x1^2*x2 - x1^2 + x1*x2^2 - x2^2, x1^3*x2^2 - 3*x1^3*x2 + 2*x1^3 + 2*x1^2*x2^3 - 5*x1^2*x2^2 + 4*x1^2*x2 - 2*x1^2 + x1*x2^4 - 3*x1*x2^3 + 5*x1*x2^2 - 2*x1*x2 - x2^4 + 2*x2^3 - x2^2, x1^3*x2^2 - 3*x1^3*x2 + 2*x1^3 + 2*x1^2*x2^3 - 5*x1^2*x2^2 + 4*x1^2*x2 - 2*x1^2 + x1*x2^4 - 3*x1*x2^3 + 4*x1*x2^2 - x1*x2 - x2^4 + 2*x2^3 - x2^2, x1^2*x2 - x1^2 + x1*x2^2 - x1*x2 + x1 - 2*x2^2 + x2, x1^4*x2^2 - 2*x1^4*x2 + x1^4 + 3*x1^3*x2^3 - 5*x1^3*x2^2 + 3*x1^3*x2 - x1^3 + 3*x1^2*x2^4 - 5*x1^2*x2^3 + 4*x1^2*x2^2 - x1^2*x2 + x1*x2^5 - 3*x1*x2^4 + 4*x1*x2^3 - 2*x1*x2^2 - x2^5 + 2*x2^4 - x2^3, x1*x2 - x1 + 1, x1*x2 - x1 + x2^2 - x2 + 1]