  We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a
matrix $A$ whose conventional and nonnegative ranks equal five, but the
nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$
can be represented as a sum of five rank-one matrices with nonnegative real
entries but not as a sum of five rank-one matrices with nonnegative entries in
$\mathcal{F}$.
