The realization space is
  [1   1   0   0   1   1                                                                                  0                                       x2*x3 - x2 - x3 + 1                 x2 - 1    1    1]
  [1   0   1   0   1   0                                                                x2*x3 - x2 - x3 + 1   x1*x3*x4 - x1*x3 - x1*x4 + x1 + x2*x3 - x2 - x3*x4 + x4   x1*x4 - x1 + x2 - x4   x1   x3]
  [0   0   0   1   1   1   -x1*x4^2 + 2*x1*x4 - x1 + x2*x3*x4 - x2*x3 - x2*x4 + x2 - x3*x4 + x3 + x4^2 - x4    x1*x4^2 - 2*x1*x4 + x1 + x2*x3 - x2 - x3 - x4^2 + 2*x4             x2*x4 - x4   x2   x4]
in the multivariate polynomial ring in 4 variables over ZZ
within the vanishing set of the ideal
Ideal with 5 generators
avoiding the zero loci of the polynomials
RingElem[x1*x4 - x2*x3 + x2 - x4, x1*x4 - x2*x3, x2 - x4, x1 - x3, x1*x4 - x1 - x2*x3 + x2 + x3 - x4, x1*x4 - x1 - x2*x3 + x3, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 - x1*x3*x4^2 + 3*x1*x3*x4 - 2*x1*x3 - x1*x4^2 + x1*x4 + x2^2*x3 - x2^2 + x2*x3^2*x4 - x2*x3^2 - 2*x2*x3*x4 + x2*x4 + x2 - x3^2*x4 + x3^2 + x3*x4^2 - x4, x4, x4 - 1, x3 - x4, x1*x4 - x1 - x2*x3 + 2*x2 + x3 - x4 - 1, x1*x4^2 - 2*x1*x4 + x1 - x2*x3*x4 + x2*x3 + 2*x2*x4 - x2 + x3*x4 - x3 - x4^2, x1*x3*x4^2 - 2*x1*x3*x4 + x1*x3 - x2*x3^2*x4 + x2*x3^2 + 2*x2*x3*x4 - x2*x3 - x2*x4 + x3^2*x4 - x3^2 - x3*x4^2 + x4, x1*x3*x4 - x1*x3 - x2*x3^2 + 2*x2*x3 - x2 + x3^2 - x3*x4 - x3 + 1, x3 + x4 - 1, x3, x3 - 1, x1 - 1, x1 - x2, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 + x1*x3*x4 - x1*x3 - 2*x1*x4^2 + 3*x1*x4 - x1 + x2^2*x3 - x2^2 + x2*x3*x4 - 3*x2*x3 - x2*x4 + 3*x2 - x3*x4 + 2*x3 + x4^2 - x4 - 1, x1*x4^2 - 2*x1*x4 + x1 - x2*x3*x4 + 2*x2*x3 + x2*x4 - 2*x2 - x3 - x4^2 + 2*x4, x1*x4^2 - 2*x1*x4 + x1 - x2^2*x3 + x2^2 + 2*x2*x3 - 2*x2 - x3 - x4^2 + 2*x4, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 + x1*x3*x4 - x1*x3 - x1*x4^2 + x1*x4 - x2^2*x3 + x2^2 + x2*x3*x4 + x2*x3 - x2*x4 - x2 - x3*x4 + x4, x1^2*x2*x4^2 - 2*x1^2*x2*x4 + x1^2*x2 - x1^2*x4^3 + 2*x1^2*x4^2 - x1^2*x4 - x1*x2^2*x3*x4 + x1*x2^2*x3 + x1*x2^2*x4 - x1*x2^2 + x1*x2*x3*x4^2 - x1*x2*x3 - 4*x1*x2*x4^2 + 6*x1*x2*x4 - 2*x1*x2 - x1*x3*x4^2 + x1*x3*x4 + 2*x1*x4^3 - 2*x1*x4^2 - x1*x4 + x1 + x2^3*x3 - x2^3 + x2^2*x3*x4 - 4*x2^2*x3 - x2^2*x4 + 4*x2^2 - x2*x3*x4^2 + 4*x2*x3 + 3*x2*x4^2 - 4*x2*x4 - 2*x2 + x3*x4^2 - x3*x4 - x3 - x4^3 + 2*x4, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 + x1*x3*x4 - x1*x3 - 2*x1*x4^2 + 3*x1*x4 - x1 + x2^2*x3 - x2^2 + x2*x3*x4 - 2*x2*x3 - x2*x4 + 2*x2 - x3*x4 + x3 + x4^2 - x4, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 + x1*x3*x4 - x1*x3 - x1*x4^2 + x1*x4 + x2^2*x3 - x2^2 - x2*x3 + x2, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + x1*x2*x4 - x1*x2 + x1*x3*x4 - x1*x3 - x1*x4^2 + x1*x4 + x2^2*x3 - x2^2 - 2*x2*x3 + 2*x2 + x3 - 1, x1 + x2 - 1, x1, x2 - 1, x2, x1*x4 - x1 + x2 - x4, x1^2*x4^3 - 3*x1^2*x4^2 + 3*x1^2*x4 - x1^2 - x1*x2*x3*x4^2 + 2*x1*x2*x3*x4 - x1*x2*x3 + x1*x2*x4^2 - 2*x1*x2*x4 + x1*x2 + x1*x3*x4^2 - 2*x1*x3*x4 + x1*x3 - 2*x1*x4^3 + 5*x1*x4^2 - 4*x1*x4 + x1 + x2^2*x3*x4 - x2^2*x4 + x2*x3*x4^2 - 4*x2*x3*x4 + x2*x3 - x2*x4^2 + 4*x2*x4 - x2 - x3*x4^2 + 3*x3*x4 - x3 + x4^3 - 2*x4^2, x1*x4 - x1 + x2*x4 - 2*x4 + 1, x1^2*x4^3 - 3*x1^2*x4^2 + 3*x1^2*x4 - x1^2 - x1*x2*x3*x4^2 + 2*x1*x2*x3*x4 - x1*x2*x3 + 3*x1*x2*x4^2 - 6*x1*x2*x4 + 3*x1*x2 + x1*x3*x4^2 - 2*x1*x3*x4 + x1*x3 - 2*x1*x4^3 + 3*x1*x4^2 - x1 - x2^2*x3*x4 + 2*x2^2*x3 + x2^2*x4 - 2*x2^2 + x2*x3*x4^2 - 3*x2*x3 - 3*x2*x4^2 + 4*x2*x4 + x2 - x3*x4^2 + x3*x4 + x3 + x4^3 - 2*x4, x1^2*x4^2 - 2*x1^2*x4 + x1^2 - x1*x2*x3*x4 + x1*x2*x3 + 3*x1*x2*x4 - 3*x1*x2 + x1*x3*x4 - x1*x3 - 2*x1*x4^2 + x1*x4 + x1 - x2^2*x3 + x2^2 + x2*x3*x4 + x2*x3 - 3*x2*x4 + x2 - x3*x4 + x4^2 + x4 - 1, x1*x3*x4 - x1*x3 + x1*x4^2 - 3*x1*x4 + 2*x1 + x2*x3 - x2 - x3*x4 - x4^2 + 3*x4 - 1, x1*x4^2 - 2*x1*x4 + x1 + x2*x3 - x2 - x3 - x4^2 + 2*x4, x1*x4^2 - 2*x1*x4 + x1 - x2*x3*x4 + x2*x4 + x3*x4 - x4^2 + x4 - 1, x1*x4^2 - 2*x1*x4 + x1 - x2*x3*x4 + 2*x2*x3 + x2*x4 - 2*x2 + x3*x4 - 2*x3 - x4^2 + x4 + 1]