The realization space is
  [1   1   0   0   1   1                                               0                                                                                   x1*x2^3 - x1*x2^2 - x2^4 + x2^3                                                     x1*x2^2 - x1*x2 - x2^3 + x2^2                                    x1*x2^2 - x2^3    1]
  [0   1   1   0   0   1                                  x1*x2^2 - x2^3                      -x1^4*x2 + 2*x1^3*x2 + 2*x1^2*x2^3 - 2*x1^2*x2^2 - x1^2*x2 - x1*x2^3 + x1*x2^2 - x2^4 + x2^3   -x1^4 + 2*x1^3 + 2*x1^2*x2^2 - 2*x1^2*x2 - x1^2 - x1*x2^2 + x1*x2 - x2^3 + x2^2    -x1^4 + x1^3 + 2*x1^2*x2^2 - x1^2*x2 - x1*x2^2   x1]
  [0   0   0   1   1   1   x1^4 - x1^3 - 2*x1^2*x2^2 + x1^2*x2 + x1*x2^2   x1^4*x2 - x1^4 - x1^3*x2^2 + x1^3 - 2*x1^2*x2^3 + 4*x1^2*x2^2 - 2*x1^2*x2 + 2*x1*x2^4 - 2*x1*x2^3 - x2^4 + x2^3    -x1^3*x2 + x1^3 + x1^2*x2 - x1^2 + 2*x1*x2^3 - 3*x1*x2^2 + x1*x2 - x2^3 + x2^2   -x1^3*x2 + x1^2*x2 + 2*x1*x2^3 - x1*x2^2 - x2^3   x2]
in the multivariate polynomial ring in 2 variables over ZZ
within the vanishing set of the ideal
Ideal with 2 generators
avoiding the zero loci of the polynomials
RingElem[x1^3 - x1^2 - x1*x2^2 + x1*x2 - x2^3 + x2^2, x1, x1 - x2, x1^5 - x1^4 - 2*x1^3*x2^2 + x1^3*x2 + x1^2*x2^2 - x1*x2^3 + x2^4, x1^5*x2 - x1^5 + x1^4 - 2*x1^3*x2^3 + 2*x1^3*x2^2 - 2*x1^3*x2 + x1^2*x2^2 + x2^5 - x2^4, x1 - 1, x1^5 - 2*x1^4 - 2*x1^3*x2^2 + x1^3*x2 + x1^3 + 3*x1^2*x2^2 - x1^2*x2 - x1*x2^3 + x2^4 - x2^3, x1^3 - x1^2 - 2*x1*x2^2 + x1*x2 + x2^2, x1^4*x2 - x1^4 + x1^3 - 2*x1^2*x2^3 + 3*x1^2*x2^2 - 2*x1^2*x2 - x2^4 + x2^3, x1 + x2 - 1, x1^2 - x1*x2 - x2^2, x1^5 - x1^4 - 2*x1^3*x2^2 + x1^2*x2^2 + x1^2*x2 + x1*x2^3 - x1*x2^2 + x2^4 - x2^3, x1^8 - 3*x1^7 - 3*x1^6*x2^2 + 2*x1^6*x2 + 3*x1^6 - x1^5*x2^3 + 9*x1^5*x2^2 - 5*x1^5*x2 - x1^5 + 2*x1^4*x2^4 - 3*x1^4*x2^3 - 5*x1^4*x2^2 + 3*x1^4*x2 + 2*x1^3*x2^5 - 7*x1^3*x2^4 + 8*x1^3*x2^3 - 2*x1^3*x2^2 - x2^7 + 2*x2^6 - x2^5, x1^5 - 2*x1^4 - 2*x1^3*x2^2 + x1^3*x2 + x1^3 + 3*x1^2*x2^2 - x1^2*x2 - x1*x2^3 - x1*x2^2 + x2^4, x1^5 - x1^4 - 2*x1^3*x2^2 + x1^3*x2 + x1^2*x2^2 - x1*x2^3 + x1*x2^2 + x2^4 - x2^3, x2 - 1, x2, x1^2 - x1 - 2*x2^2 + 2*x2, x1^5*x2 - x1^5 + x1^4 - 2*x1^3*x2^3 + 2*x1^3*x2^2 - 2*x1^3*x2 + x1^2*x2^3 - x1*x2^4 + x1*x2^3 + x2^5 - x2^4, x1^7*x2 - x1^7 - x1^6*x2 + 2*x1^6 - 4*x1^5*x2^3 + 5*x1^5*x2^2 - 3*x1^5*x2 - x1^5 + 3*x1^4*x2^3 - 3*x1^4*x2^2 + 3*x1^4*x2 + 4*x1^3*x2^5 - 6*x1^3*x2^4 + 4*x1^3*x2^3 - 3*x1^3*x2^2 - x1^2*x2^5 - x1^2*x2^4 + 2*x1^2*x2^3 - 2*x2^7 + 4*x2^6 - 2*x2^5, x1^5 - x1^4 - 2*x1^3*x2^2 + x1^3*x2 + 2*x1^2*x2^2 - 2*x1*x2^3 + x2^4, x1^7 - 2*x1^6 - 4*x1^5*x2^2 + 2*x1^5*x2 + x1^5 + 7*x1^4*x2^2 - 2*x1^4*x2 + 4*x1^3*x2^4 - 6*x1^3*x2^3 - 2*x1^3*x2^2 - 5*x1^2*x2^4 + 5*x1^2*x2^3 + 4*x1*x2^5 - 2*x1*x2^4 - 2*x2^6 + x2^5, x1^4 - x1^3 - 2*x1^2*x2^2 + x1^2*x2 + 2*x1*x2^2 - x2^3, x1^3 + x1^2*x2 - x1^2 - 2*x1*x2^2 - 2*x2^3 + 3*x2^2, x1^4 - 2*x1^3 - 2*x1^2*x2^2 + 2*x1^2*x2 + x1^2 + x1*x2^2 - x1*x2 + x2^3 - x2^2, x1^5 - 2*x1^4 - 2*x1^3*x2^2 + 2*x1^3*x2 + x1^3 + x1^2*x2^2 - x1^2*x2 + x2^4 - x2^3, x1^7 - 3*x1^6 - 4*x1^5*x2^2 + 3*x1^5*x2 + 3*x1^5 + 8*x1^4*x2^2 - 5*x1^4*x2 - x1^4 + 4*x1^3*x2^4 - 6*x1^3*x2^3 - 3*x1^3*x2^2 + 2*x1^3*x2 - 3*x1^2*x2^4 + 4*x1^2*x2^3 - x1*x2^4 + x1*x2^3 - 2*x2^6 + 5*x2^5 - 3*x2^4, x1^4 + x1^3*x2 - 3*x1^3 - 2*x1^2*x2^2 + x1^2*x2 + 2*x1^2 - 2*x1*x2^3 + 5*x1*x2^2 - 3*x1*x2 + x2^3 - x2^2, x1^5 - 2*x1^4 - 2*x1^3*x2^2 + 2*x1^3*x2 + x1^3 + 3*x1^2*x2^2 - 3*x1^2*x2 - 2*x1*x2^3 + 2*x1*x2^2 + x2^4 - x2^3, x1^7 - 3*x1^6 - 4*x1^5*x2^2 + 3*x1^5*x2 + 3*x1^5 + 10*x1^4*x2^2 - 7*x1^4*x2 - x1^4 + 4*x1^3*x2^4 - 8*x1^3*x2^3 - 3*x1^3*x2^2 + 4*x1^3*x2 - 7*x1^2*x2^4 + 12*x1^2*x2^3 - 4*x1^2*x2^2 + 4*x1*x2^5 - 5*x1*x2^4 + x1*x2^3 - 2*x2^6 + 3*x2^5 - x2^4, 2*x1^3*x2 - x1^3 - x1^2*x2^2 - 2*x1^2*x2 + x1^2 - 4*x1*x2^3 + 6*x1*x2^2 - x1*x2 + 2*x2^4 - x2^3 - x2^2, x1^2 - x1 - 2*x2^2 + x2, x1^4 + x1^3*x2^2 - 2*x1^3*x2 - x1^3 - 2*x1^2*x2^2 + 3*x1^2*x2 - 2*x1*x2^4 + 4*x1*x2^3 - 2*x1*x2^2 + x2^4 - x2^3, x1^4 - x1^3 - 2*x1^2*x2^2 + x1^2*x2 + x2^3]