  The enhanced principal rank characteristic sequence (epr-sequence) of an $n
\times n$ symmetric matrix over a field $\mathbb{F}$ was recently defined as
$\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or
$\tt N$ based on whether all, some (but not all), or none of the order-$k$
principal minors of the matrix are nonzero. Here, a complete characterization
of the epr-sequences that are attainable by symmetric matrices over the field
$\mathbb{Z}_2$, the integers modulo $2$, is established. Contrary to the
attainable epr-sequences over a field of characteristic $0$, our
characterization reveals that the attainable epr-sequences over $\mathbb{Z}_2$
possess very special structures. For more general fields of characteristic $2$,
some restrictions on attainable epr-sequences are obtained.
