Isomorphisms of section spaces

The space of real (resp. complex) sections of a vector bundle $E \to M$ is a real (resp. complex) $C^\infty(M)$-module. One has the following isomorphisms of modules:

1. \[ \Gamma(E \otimes F) = \Gamma(E) \otimes \Gamma(F) \]

This is trivial locally, and follows globally by taking a partition of unity. This is very false in the algebraic category.