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