  We investigate Poincar\'e series, where we average products of terms of
Fourier series of real-analytic Siegel modular forms. There are some (trivial)
special cases for which the products of terms of Fourier series of elliptic
modular forms and harmonic Maass forms are almost holomorphic, in which case
the corresponding Poincar\'e series are almost holomorphic as well. In general
this is not the case. The main point of this paper is the study of
Siegel-Poincar\'e series of degree $2$ attached to products of terms of Fourier
series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We
establish conditions on the convergence and nonvanishing of such
Siegel-Poincar\'e series. We surprisingly discover that these Poincar\'e series
are almost holomorphic Siegel modular forms, although the product of terms of
Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular
forms (in contrast to the elliptic case) is not almost holomorphic. Our proof
employs tools from representation theory. In particular, we determine some
constituents of the tensor product of Harish-Chandra modules with walls.
