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