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