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