Consider the hamiltonian $$ H = - \frac{1}{2} \nabla^2 + V. $$ The potential $V : (\mathbb{R^3})^N \to \mathbb{R}$ is symmetric, so for each eigenvalue, there is an antisymmetric eigenvector. There is not any spin operator inside $V$. It can be assumed that the hamiltonian describes a fermionic many-body problem.
Suppose I give you $2^N$ functions $f_i : (\mathbb{R}^{3})^N \to \mathbb{C}$ for $i = 1,2,3,\cdots,2^N$ that are square-integrable and eigenvectors of $H$ with eigenvalue $E_i$. If $\sigma_j \in \{1/2,-1/2\}$ for each $j \in \{1,2,3,\cdots, 2^N\}$; how would you find out which $f_i$ corresponds to the spin configuration $\sigma_1\sigma_2\cdots\sigma_N$?