The realization space is
  [1   1   0   0   1   1    0         x1      1   x1^2 - x1 - 1      x1^2 - x1 - 1]
  [1   0   1   0   1   0   x1       x1^3   x1^2            x1^4        x1^4 - x1^3]
  [0   0   0   1   1   1    1   x1^2 - 1     x1       x1^3 - x1   x1^3 - x1^2 - x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^12 - 3*x1^11 + 2*x1^10 + 3*x1^9 - 5*x1^8 + 3*x1^6 - x1^5 - x1^4)
avoiding the zero loci of the polynomials
RingElem[x1 + 1, x1, x1^3 - x1^2 + 1, x1 - 1, x1^4 - x1^3 - x1^2 + x1 + 1, x1^3 - 2*x1^2 + x1 + 1, x1^3 - x1 + 1, x1^2 + x1 - 1, x1^2 - x1 - 1]