The realization space is
  [1   1   0   0   1   1                             0                           x3*x4      x4    1    1]
  [0   1   1   0   0   1                         x3*x4                         x2*x3^2   x2*x3   x1   x3]
  [0   0   0   1   1   1   -x1*x4^2 + x2*x3*x4 - x2*x3   -x1*x4^2 + 2*x2*x3*x4 - x2*x3   x2*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, x1*x4 - x2*x3 + x2 - x4, x2 - x4, x1 - x3, x1*x4 - x1 - x2*x3 + x3, x1*x4 - x1 - x2*x3 + x2 + x3 - x4, x1^2*x4^2 - x1*x2*x3*x4 + x1*x2*x3 - x1*x3*x4^2 + x2*x3^2*x4 - x2*x3^2 + x2*x3*x4 - x3*x4^2, x3, x3 - x4, x1*x4^2 - x2*x3*x4 + x2*x3 + x4^2, x1*x4^2 - x2*x3*x4 + x2*x3, x1*x3*x4^2 - x1*x4^2 - x2*x3^2*x4 + x2*x3^2 + 2*x2*x3*x4 - x2*x3 - x3*x4^2, x1*x4^2 - 2*x2*x3*x4 + x2*x3 + x3*x4^2, x4, x1*x4 - x2*x3 + x3, x1*x3*x4^2 - x1*x4^2 - x2*x3^2*x4 + 2*x2*x3*x4 - x2*x3 + x3^2*x4 - x3*x4^2, x1*x2*x3*x4^2 - x1*x3*x4^3 - x1*x4^3 - x2^2*x3^2*x4 + x2^2*x3^2 + x2*x3^2*x4^2 - x2*x3^2*x4 + 2*x2*x3*x4^2 - x2*x3*x4 - x3*x4^3, x1*x3*x4^2 - x1*x4^2 - x2*x3^2*x4 + x2*x3^2 + x2*x3*x4 - x2*x3 + x3*x4^2, x1*x4^2 - x2*x3*x4 + x2*x3 + x4^2 - x4, x1*x3*x4^2 - x1*x4^2 - x2*x3^2*x4 + x2*x3^2 + x2*x3*x4 - x2*x3 + x3*x4^2 - x3*x4, x3 - 1, x4 - 1, x3 + x4 - 1, x2, x2 - 1, x1^2*x4^2 - 2*x1*x2*x3*x4 + x1*x2*x3 + x2^2*x3^2, x1^2*x4^2 - 2*x1*x2*x3*x4 + x1*x2*x3 - x1*x4^2 + x2^2*x3^2 + x2*x3*x4 - x2*x3, x1^2*x4^2 - 2*x1*x2*x3*x4 + x1*x2*x3 + x1*x3*x4 + x2^2*x3^2 - x2*x3^2, x1^2*x4^2 - 2*x1*x2*x3*x4 + x1*x2*x3 + x1*x3*x4 - x1*x4^2 + x2^2*x3^2 - x2*x3^2 + x2*x3*x4 - x2*x3, x1*x4 - x2*x3 + x4, x1^2*x4^2 - x1*x2*x3*x4 + x1*x2*x3 + x2*x3*x4, x1^2*x4^2 - x1*x2*x3*x4 + x1*x2*x3 - x1*x4^2 + 2*x2*x3*x4 - x2*x3, x1^2*x4^2 - x1*x2*x3*x4 + x1*x2*x3 - x1*x4^2 + 2*x2*x3*x4 - x2*x3 - x3*x4, x1 - x2, x1 - 1, x1, x1 + x2 - 1, x2*x3 - x4, x1*x2*x4^2 - x2^2*x3*x4 + x2^2*x3 + x2*x4^2 - x4^2, x1*x2*x3*x4^2 - x1*x4^3 - x2^2*x3^2*x4 + x2^2*x3^2 + 2*x2*x3*x4^2 - x2*x3*x4 - x3*x4^2, x2*x3 + x2*x4 - x4, x1*x2*x3*x4^2 - 2*x1*x4^3 - x2^2*x3^2*x4 + x2^2*x3^2 + 3*x2*x3*x4^2 - 2*x2*x3*x4, x1*x2*x3*x4^2 - 2*x1*x4^3 - x2^2*x3^2*x4 + x2^2*x3^2 + 3*x2*x3*x4^2 - 2*x2*x3*x4 - x3*x4^2, x1*x4^2 + x2*x3^2 - 2*x2*x3*x4 + x2*x3, x1*x4^2 - 2*x2*x3*x4 + x2*x3 + x3*x4, x1*x4^2 - x2*x3^2 - 2*x2*x3*x4 + x2*x3 + x3*x4, x1*x4^2 - 2*x2*x3*x4 + x2*x3, x1*x4^2 - x2*x3*x4 + x2*x3 + x3*x4, x1*x4^2 - x2*x3*x4 + x2*x3 - x3*x4]