  We study the half-BPS circular Wilson loop in ${\cal N}=4$ super Yang-Mills
with orthogonal gauge group. By supersymmetric localization, its expectation
value can be computed exactly from a matrix integral over the Lie algebra of
$SO(N)$. We focus on the large $N$ limit and present some simple quantitative
tests of the duality with type IIB string theory in $AdS_5\times
\mathbb{RP}^5$. In particular, we show that the strong coupling limit of the
expectation value of the Wilson loop in the spinor representation of the gauge
group precisely matches the classical action of the dual string theory object,
which is expected to be a D5-brane wrapping a $\mathbb{RP}^4$ subspace of
$\mathbb{RP}^5$. We also briefly discuss the large $N$, large $\lambda$ limits
of the $SO(N)$ Wilson loop in the symmetric/antisymmetric representations and
their D3/D5-brane duals. Finally, we use the D5-brane description to extract
the leading strong coupling behavior of the "bremsstrahlung function"
associated to a spinor probe charge, or equivalently the normalization of the
two-point function of the displacement operator on the spinor Wilson loop, and
obtain agreement with the localization prediction.
