Abstract: We introduce Gauss-Berezin operators that are super-hybrids of the usual Gaussian integral operators and Berezin operators in Grassmann algebra. These operators are enumerated by points of the Lagrangian super-Grassmannian. We show that product of Gauss-Berezin operators of general position is an operator of the same type. In particular, this 'almost semigroup' of Gauss--Berezin operators contains the supergroup OSp.