  A minor of a matrix is quasi-principal if it is a principal or an
almost-principal minor. The quasi principal rank characteristic sequence
(qpr-sequence) of an $n\times n$ symmetric matrix is introduced, which is
defined as $q_1 q_2 \cdots q_n$, where $q_k$ is $\tt A$, $\tt S$, or $\tt N$,
according as all, some but not all, or none of its quasi-principal minors of
order $k$ are nonzero. This sequence extends the principal rank characteristic
sequences in the literature, which only depend on the principal minors of the
matrix. A necessary condition for the attainability of a qpr-sequence is
established. Using probabilistic techniques, a complete characterization of the
qpr-sequences that are attainable by symmetric matrices over fields of
characteristic $0$ is given.
