The realization space is
  [1   1   0   0   1   1                       0         x1 - 1                  x1^2*x2 - x1^2 - x1*x2^2 - x1*x2 + 2*x1 + x2^2 - 1                        x1^2 - x1*x2 - 2*x1 + x2 + 1    1]
  [1   0   1   0   1   0   x1*x2 - x1 - x2^2 + 1   x1*x2 - x2^2                      -x1^2 - x1*x2^3 + x1*x2 + 2*x1 + x2^3 - x2 - 1      x1^2 - 2*x1*x2 - 2*x1 - x2^3 + x2^2 + 2*x2 + 1   x1]
  [0   0   0   1   1   1              x1*x2 - x2     x1*x2 - x2   x1^2*x2 - x1^2 - 2*x1*x2^2 + 2*x1 - x2^4 + 2*x2^3 + x2^2 - x2 - 1   x1^2 + x1*x2^2 - 2*x1*x2 - 2*x1 - x2^3 + 2*x2 + 1   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 - 1, x1*x2 - x1 - 2*x2^2 + 1, x1^3*x2 - x1^3 - 3*x1^2*x2^2 + 2*x1^2 + 2*x1*x2^3 + 2*x1*x2^2 - x1*x2 - x1 - x2^4, x1 - x2 - 1, x2 - 1, x1^2*x2 - x1^2 - 2*x1*x2^2 + x1 + x2^3, x1 + x2^2 - x2 - 1, x1^2 - x1*x2 - x1 + x2^2, x2, x1^4*x2^2 - x1^4*x2 - 3*x1^3*x2^3 + x1^3*x2^2 + x1^3 + 4*x1^2*x2^4 - 2*x1^2*x2^3 + x1^2*x2^2 + 3*x1^2*x2 - 3*x1^2 - 3*x1*x2^5 + x1*x2^4 + 4*x1*x2^3 - 6*x1*x2^2 - 2*x1*x2 + 3*x1 + x2^6 - 2*x2^4 + 3*x2^2 - 1, x1^5*x2^3 - 2*x1^5*x2^2 + 2*x1^5*x2 - x1^5 - 4*x1^4*x2^4 + 6*x1^4*x2^3 - 3*x1^4*x2^2 - 4*x1^4*x2 + 4*x1^4 + 7*x1^3*x2^5 - 8*x1^3*x2^4 - 3*x1^3*x2^3 + 14*x1^3*x2^2 - 6*x1^3 - 7*x1^2*x2^6 + 6*x1^2*x2^5 + 9*x1^2*x2^4 - 12*x1^2*x2^3 - 10*x1^2*x2^2 + 4*x1^2*x2 + 4*x1^2 + 4*x1*x2^7 - 2*x1*x2^6 - 9*x1*x2^5 + 6*x1*x2^4 + 8*x1*x2^3 - 2*x1*x2 - x1 - x2^8 + 3*x2^6 - 4*x2^4 + x2^2, x1^3*x2 - 2*x1^2*x2^2 - x1^2 + 2*x1*x2^3 - x1*x2 + 2*x1 - x2^4 + x2^2 - 1, x1^2 - x1*x2 + x2^2 - 1, x1 - x2, x1^3*x2^2 - x1^3*x2 + x1^3 - 2*x1^2*x2^3 + 2*x1^2*x2^2 - 2*x1^2 + 2*x1*x2^4 - 2*x1*x2^3 - x1*x2^2 + x1*x2 + x1 - x2^5 + x2^4 + x2^3 - x2^2, x1^2*x2 - x1*x2^2 - x1 + x2^3 - x2^2 + 1, x1^2*x2 - x1*x2^2 + x1*x2 - x1 + x2^3 - x2^2 - x2 + 1, x1 + x2 - 1, x1, x1^4*x2^4 - 2*x1^4*x2^3 + 4*x1^4*x2^2 - 3*x1^4*x2 + x1^4 - 3*x1^3*x2^5 + 7*x1^3*x2^4 - 10*x1^3*x2^3 - 2*x1^3*x2^2 + 8*x1^3*x2 - 4*x1^3 + 4*x1^2*x2^6 - 9*x1^2*x2^5 + 3*x1^2*x2^4 + 20*x1^2*x2^3 - 12*x1^2*x2^2 - 6*x1^2*x2 + 6*x1^2 - 3*x1*x2^7 + 5*x1*x2^6 + 6*x1*x2^5 - 16*x1*x2^4 - 6*x1*x2^3 + 14*x1*x2^2 - 4*x1 + x2^8 - x2^7 - 3*x2^6 + 2*x2^5 + 6*x2^4 - 2*x2^3 - 4*x2^2 + x2 + 1, x1^3*x2^3 - x1^3*x2^2 + 2*x1^3*x2 - x1^3 - 2*x1^2*x2^4 + 3*x1^2*x2^3 - 4*x1^2*x2^2 - 3*x1^2*x2 + 3*x1^2 + 2*x1*x2^5 - 3*x1*x2^4 - x1*x2^3 + 8*x1*x2^2 - 3*x1 - x2^6 + x2^5 + 2*x2^4 - 2*x2^3 - 3*x2^2 + x2 + 1, x1^2*x2^2 - x1^2*x2 + x1^2 - 2*x1*x2^3 + 2*x1*x2^2 - 2*x1 + x2^4 - 2*x2^2 + x2 + 1, x1^4*x2^4 - 2*x1^4*x2^3 + 3*x1^4*x2^2 - 2*x1^4*x2 + x1^4 - 3*x1^3*x2^5 + 6*x1^3*x2^4 - 6*x1^3*x2^3 - 2*x1^3*x2^2 + 5*x1^3*x2 - 4*x1^3 + 4*x1^2*x2^6 - 7*x1^2*x2^5 + 14*x1^2*x2^3 - 8*x1^2*x2^2 - 3*x1^2*x2 + 6*x1^2 - 3*x1*x2^7 + 4*x1*x2^6 + 5*x1*x2^5 - 10*x1*x2^4 - 5*x1*x2^3 + 10*x1*x2^2 - x1*x2 - 4*x1 + x2^8 - x2^7 - 2*x2^6 + x2^5 + 4*x2^4 - x2^3 - 3*x2^2 + x2 + 1, x1^3*x2^3 - 2*x1^3*x2^2 + 3*x1^3*x2 - x1^3 - 2*x1^2*x2^4 + 4*x1^2*x2^3 - 2*x1^2*x2^2 - 6*x1^2*x2 + 3*x1^2 + 2*x1*x2^5 - 3*x1*x2^4 - 3*x1*x2^3 + 7*x1*x2^2 + 3*x1*x2 - 3*x1 - x2^6 + x2^5 + 2*x2^4 - x2^3 - 3*x2^2 + 1, x1^3*x2^2 - x1^3*x2 + x1^3 - 2*x1^2*x2^3 + x1^2*x2^2 + x1^2*x2 - 3*x1^2 + 2*x1*x2^4 - 3*x1*x2^2 + x1*x2 + 3*x1 - x2^5 + x2^3 + x2^2 - x2 - 1, x1^3*x2^2 - 2*x1^2*x2^3 + x1^2*x2^2 - 3*x1^2*x2 + x1^2 + 2*x1*x2^4 - 2*x1*x2^3 + x1*x2^2 + 4*x1*x2 - 2*x1 - x2^5 + x2^4 + x2^3 - 2*x2^2 - x2 + 1, x1^2*x2 - 3*x1*x2 + x1 - x2^3 + 2*x2^2 + x2 - 1, x1*x2^2 - x1*x2 + x1 - x2^3 + x2^2 - 1, 2*x1^2*x2^2 - 2*x1^2*x2 + x1^2 - 3*x1*x2^3 + x1*x2^2 + 2*x1*x2 - 2*x1 + x2^4 + x2^3 - 2*x2^2 + 1, 2*x1*x2 - x1 - x2^2 - x2 + 1]