The realization space is
  [1   0   1   0   1    0                                                           x2^2 - x2    1                                                           x2^2 - x2                                                  x2^2 - x2                                                  x2^2 - x2]
  [0   1   1   0   0    1                                                           x2^2 - x2   x1                                                       -x1*x2 + x2^2                                            x1*x2^2 - x1*x2   -x1^3 + x1^2*x2 + x1^2 + 2*x1*x2^2 - 3*x1*x2 - x2^2 + x2]
  [0   0   0   1   1   -1   -x1^3 + x1^2*x2 + x1^2 + 2*x1*x2^2 - 2*x1*x2 + x2^3 - 3*x2^2 + x2   x2   -x1^3 + x1^2*x2 + x1^2 + 2*x1*x2^2 - 2*x1*x2 + x2^3 - 3*x2^2 + x2   -x1^3 + x1^2*x2 + x1^2 + x1*x2^2 - x1*x2 + x2^3 - 2*x2^2                                                x2^3 - x2^2]
in the multivariate polynomial ring in 2 variables over ZZ
within the vanishing set of the ideal
Ideal with 3 generators
avoiding the zero loci of the polynomials
RingElem[x1 - 1, x1^2 - x1*x2 - x2^2, x1^5 - 2*x1^4*x2 - x1^4 - 2*x1^3*x2^2 + 4*x1^3*x2 + 2*x1^2*x2^3 + x1^2*x2^2 - 2*x1^2*x2 + 3*x1*x2^4 - 6*x1*x2^3 + 2*x1*x2^2 + x2^5 - 4*x2^4 + 3*x2^3, x1^2 - x1*x2 - x2^2 + x2, x1^5 - 2*x1^4*x2 - x1^4 - 2*x1^3*x2^2 + 4*x1^3*x2 + 2*x1^2*x2^3 + x1^2*x2^2 - 2*x1^2*x2 + 3*x1*x2^4 - 6*x1*x2^3 + 2*x1*x2^2 + x2^5 - 4*x2^4 + 4*x2^3 - x2^2, x1^2 - x1*x2 - 2*x2^2 + x2, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 3*x1*x2 - x2^3 + 2*x2^2 - x2, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 3*x1*x2 - x2^3 + 3*x2^2 - 2*x2, x2, x2 - 1, x1^5 - 2*x1^4*x2 - x1^4 - 3*x1^3*x2^2 + 5*x1^3*x2 + 3*x1^2*x2^3 - 2*x1^2*x2 + 5*x1*x2^4 - 10*x1*x2^3 + 4*x1*x2^2 + 2*x2^5 - 6*x2^4 + 5*x2^3 - x2^2, x1^2 - x1*x2 - 2*x2^2 + 2*x2, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 2*x1*x2 - x2^3 + 3*x2^2 - x2, x1^5 - 2*x1^4*x2 - x1^4 - 3*x1^3*x2^2 + 5*x1^3*x2 + 3*x1^2*x2^3 + x1^2*x2^2 - 3*x1^2*x2 + 5*x1*x2^4 - 11*x1*x2^3 + 5*x1*x2^2 + 2*x2^5 - 8*x2^4 + 9*x2^3 - 3*x2^2, x1^2 - 2*x1*x2 + x2, x1 + x2 - 1, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 3*x1*x2 - x1 - x2^3 + 2*x2^2, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 3*x1*x2 - x1 - x2^3 + 3*x2^2 - 2*x2 + 1, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 3*x1*x2 - x1 - x2^3 + 3*x2^2 - x2, x1, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 2*x1*x2 - x2^3 + 2*x2^2, x1^3 - x1^2*x2 - x1^2 - x1*x2^2 + x1*x2 - x2^3 + 3*x2^2 - x2, x1^3 - x1^2*x2 - x1^2 - x1*x2^2 + x1*x2 - x2^3 + 2*x2^2, x1^3 - x1^2*x2 - 2*x1*x2^2 + x1*x2 - x2^3, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 2*x1*x2 - x2^3 + 2*x2^2 - x2, x1^3 - x1^2*x2 - x1^2 - 2*x1*x2^2 + 2*x1*x2 - x2^3 + 4*x2^2 - 2*x2, x1 - x2, x1^3 - x1^2*x2 - 2*x1*x2^2 + x1*x2 - x2^3 + 2*x2^2 - x2, x1 + x2]