The realization space is
  [1   1   0   0   1   1        0        2         2         2*x1 + 2          2*x1 + 2]
  [1   0   1   0   1   0   x1 - 1   x1 + 1   -x1 + 1        -x1^2 + 1   x1^2 - 2*x1 + 1]
  [0   0   0   1   1   1     2*x1     2*x1     -2*x1   -2*x1^2 + 2*x1     2*x1^2 + 2*x1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (-32*x1^6 + 48*x1^5 - 32*x1^3 + 32*x1^2 - 16*x1)
avoiding the zero loci of the polynomials
RingElem[x1, 2, x1 - 1, 3*x1^2 + 1, 3*x1 + 1, x1 + 1, x1^2 - 4*x1 - 1, x1^2 + 4*x1 - 1, x1^2 + 1, 3*x1 - 1]