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

Gauss--Berezin integral operators and spinors over supergroups $\mathrm{OSp}(2p|2q)$, and Lagrangian super-Grasmannians