There is a well-known in index theory "difference bundle" construction of Atiyah( see for example the original paper "Clifford modules"). And also there is a corresponding formula for the tensor product of two " difference bundles" representing it as a new " difference bundle".

However the proof ( at least the original proof of Atiyah) of the tensor product formula is quite indirect and non-intuitive. I wonder if there is a "straightforward bare-hands" proof of this seemingly not very complicated formula. It would be interesting to see such a proof even assuming that all bundles under consideration are trivial.