  We obtain explicit formulas for the spinor representation $\rho$ of the real
orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral
'Gauss--Berezin' operators. Next, we extend $\rho$ to a complex domain and get
a representation of a larger semigroup, which is a counterpart of Olshanski
subsemigroups in semisimple Lie groups. Further, we show that $\rho$ can be
extended to an operator-valued function on a certain domain in the Lagrangian
super-Grassmannian (graphs of elements of the supergroup
$\mathrm{OSp}(2p|2q,\mathbb{C})$ are Lagrangian super-subspaces) and show that
this function is a 'representation' in the following sense: we consider
Lagrangian subspaces as linear relations and composition of two Lagrangian
relations in general position corresponds to a product of Gauss--Berezin
operators
