The realization space is
  [1   x1 + 1   1   1   0    1   0      x1^2   x1^2 + x1      0     x1^2 + x1]
  [1       x1   0   0   1   x1   0      x1^2        x1^2   x1^2   x1^3 + x1^2]
  [1       x1   0   1   0    0   1   -x1 + 1           1      1             1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^12 + 3*x1^11 + x1^10 - 3*x1^9 + 3*x1^7 - x1^5)
avoiding the zero loci of the polynomials
RingElem[x1^3 + 2*x1^2 - x1 - 1, x1^2 + x1 - 1, x1, x1 + 2, x1 + 1, x1 - 1, x1^4 + x1^3 - x1^2 - x1 + 1, x1^3 + x1^2 - 1, x1^3 - x1 + 1]