The realization space is
  [1   0   1                1   0   1    0   x1^2 + x1 - 1                1       x1^2 + x1 - 1                  1]
  [0   1   1   -x1^2 - x1 + 1   0   0    1   x1^2 + x1 - 1   -x1^2 - x1 + 1   -x1^3 - x1^2 + x1   x1^3 + x1^2 - x1]
  [0   0   0                0   1   1   -1            x1^2               x1                x1^2                 x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (-x1^7 - 2*x1^6 + x1^5 + 3*x1^4 - x1^2)
avoiding the zero loci of the polynomials
RingElem[x1, x1 + 1, x1 - 1, x1^3 + x1^2 - 1, x1^2 + x1 - 1, x1^3 + 2*x1^2 + x1 - 1, x1^4 + x1^3 - x1^2 - x1 + 1, 2*x1 - 1]