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 semi-simple 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.