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