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