The realization space is
  [1   1   0   0   1   1               0   x1^4 + x1^3 - 2*x1^2 - x1 + 1    x1^2 - 1   x1^4 + x1^3 - 2*x1^2 - x1 + 1           x1^2 - 1]
  [0   1   1   0   0   1   x1^2 + x1 - 1               -x1^3 - x1^2 + x1         -x1              x1^4 + x1^3 - x1^2   x1^3 + x1^2 - x1]
  [0   0   0   1   1   1      -x1^3 + x1                     x1^4 - x1^2   x1^3 - x1                     x1^4 - x1^2          x1^3 - x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (-x1^12 - x1^11 + 5*x1^10 + 3*x1^9 - 12*x1^8 - 3*x1^7 + 16*x1^6 - 11*x1^4 + 2*x1^3 + 3*x1^2 - x1)
avoiding the zero loci of the polynomials
RingElem[x1, x1 + 1, x1 - 1, 2*x1^3 - 2*x1 + 1, x1^2 + x1 - 1, x1^4 + x1^3 - x1^2 - x1 + 1, x1^3 - x1 + 1, x1^4 + x1 - 1, x1^3 - x1^2 - 2*x1 + 1, x1^3 + x1^2 - 1]