Let
X
X
be a geometrically connected smooth projective curve of genus
g
X
≥
2
g_X \geq 2
over
R
\mathbb {R}
. Let
M
(
r
,
ξ
)
M(r, \xi )
be the coarse moduli space of geometrically stable vector bundles
E
E
over
X
X
of rank
r
r
and determinant
ξ
\xi
, where
ξ
\xi
is a real point of the Picard variety
P
i
c
_
d
(
X
)
\underline {\mathrm {Pic}}^d( X)
. If
g
X
=
r
=
2
g_X = r = 2
, then let
d
d
be odd. We compute the Brauer group of
M
(
r
,
ξ
)
M(r,\xi )
.