  In the first part of the paper we survey several results from Popa's
deformation/rigidity theory on the classification of tensor product
decompositions of large natural classes of II$_1$ factors. Using a m\'elange of
techniques from deformation/rigidity theory, model theory, and the recent works
\cite{CIOS21,CDI22} we highlight an uncountable family of existentially closed
II$_1$ factors $M$ which do not admit tensor product decompositions $M= P\bar
\otimes Q$ into diffuse factors where $Q$ is full. In the last section we
discuss several open problems regarding the structural theory of existentially
closed factors.
