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