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