Standard definition for the spin coherent state (CSS) for the system of $N$ identical particles reads
$$ |\theta, \phi\rangle = \bigotimes\limits_{k=1}^{N} \left[ \cos\frac{\theta}{2} |0\rangle_k + e^{i \phi} \sin \frac{\theta}{2} |1 \rangle_k \right], $$ where $|0 \rangle$ and $|1 \rangle$ are the eigenvectors of the Pauli matrix $\sigma_z^{(k)}$ on the $k$-th side. For the spinless fermions, the dimension of the Hilbert space is $\deg \mathcal{H} = 2^N$. However, for the case of electrons with spin, it is $4^N$ and one needs 4 vectors instead of two above. Something like $\{ |0\rangle_{k \uparrow}, |1\rangle_{k \uparrow}|0\rangle_{k \downarrow}|1\rangle_{k \downarrow} \}$. I wonder how the expression for $|\theta, \phi \rangle$ modifies in the case of electrons with spin. Should it be just additional summation over spin of not?
