The realization space is
  [1   0   1   1    1            x2*x3 - x3   0   x2 - 1                               0      x2 - 1    1]
  [0   1   1   1   x1   x2^2*x3 + x2^2 - x2   0        0                          x2 - 1   x2^2 - x2   x2]
  [0   0   0   1   x1            x2*x3 - x3   1      -x3   x2^2*x3 - x2*x3 + x2 + x3 - 1         -x3   x3]
in the multivariate polynomial ring in 3 variables over ZZ
within the vanishing set of the ideal
Ideal with 6 generators
avoiding the zero loci of the polynomials
RingElem[x2, x3, x2 - 1, x1 - x2, x2 + x3 - 1, x2^2*x3 - x2*x3 + x2 + x3 - 1, x2*x3 + 1, x2*x3 - x3 + 1, x1*x2^2*x3 - x1*x2*x3 + x1*x3 - x2^3*x3 + x2^2*x3 - x2^2 + x2 - x3, x2^4*x3 + x2^3*x3^2 - 2*x2^3*x3 + x2^3 + 2*x2^2*x3 - 2*x2^2 - x2*x3^2 + x2 + x3^2 - x3, x1*x2 - x1*x3 - x1 + x3, x3 - 1, x2^2*x3 + x2^2 - x2*x3 - x2 + x3, x1*x2^2 + x1*x2*x3^2 - x1*x2 - x1*x3^2 + x1*x3 - x2^2*x3^2, x1, x2 - x3, x1 - x3, x1 - 1, x2^3*x3 - x2^2*x3 + x2^2 + x2*x3 - x2 + x3, x2^3*x3 - 2*x2^2*x3 + x2^2 + 2*x2*x3 - 2*x2 + 1, x2^2*x3 - 2*x2*x3 + x2 + 2*x3 - 1, x1*x2^2*x3 - x1*x2*x3 + x1*x3 - x2^3*x3 + x2^2*x3 - x2^2 - x2*x3 + x2 - x3, x1*x2 - x1 + x3, x2^2 - x2 + 1, x1*x2^2 - x1*x2*x3 - x1*x2 + x1*x3 + x2^2*x3, x2^2 - x2 + x3, x1*x2^2 - 2*x1*x2 + x1*x3 + x1 - x3, x2^4*x3^2 + x2^4*x3 - x2^3*x3^2 - x2^3*x3 + x2^3 + x2^2*x3^2 - 2*x2^2 - x2*x3^2 + x2*x3 + x2 + x3^2 - x3, x2^4*x3^2 + x2^4*x3 - x2^3*x3^2 - x2^3*x3 + x2^3 + x2^2*x3^2 - 2*x2^2 + x2*x3 + x2 - x3, x2^4*x3^2 + x2^4*x3 - 2*x2^3*x3^2 - x2^3*x3 + x2^3 + 3*x2^2*x3^2 - x2^2*x3 - 2*x2^2 - 2*x2*x3^2 + 3*x2*x3 + x2 + x3^2 - 2*x3, x1*x2^3*x3^2 - 2*x1*x2^2*x3^2 + 2*x1*x2*x3^2 - x1*x3^2 - x2^4*x3^2 - x2^4*x3 + x2^3*x3^2 + x2^3*x3 - x2^3 - x2^2*x3^2 + 2*x2^2 - x2*x3 - x2 + x3, x1*x2^2*x3 - x1*x2*x3 + x1*x3 - x2^2*x3 + x2*x3 - x2 - x3 + 1, x2*x3 + x2 - 1, x1*x2^3*x3 + x1*x2^3 - 2*x1*x2^2*x3 - 2*x1*x2^2 - x1*x2*x3^2 + 2*x1*x2*x3 + x1*x2 + x1*x3^2 - x1*x3 + x2^2*x3^2 + x2^2*x3 - x2*x3, x1*x2*x3 - x1*x3 - x2^2*x3 - x2^2 + x2]