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In the first part of the paper we survey several results from Popa'sdeformation/rigidity theory on the classification of tensor productdecompositions of large natural classes of II$_1$ factors. Using a m\'elange oftechniques from deformation/rigidity theory, model theory, and the recent works\cite{CIOS21,CDI22} we highlight an uncountable family of existentially closedII$_1$ 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 wediscuss several open problems regarding the structural theory of existentiallyclosed factors.

Tensor product indecomposability results for existentially closed factors