Multiplication in the wavelet domain, what does it look like in real space?
They can be  used for time-varying filtering of non-stationary signals, under names like Gabor multipliers, STFT filters for instance.  You can find a review reference in the chapter Linear Time-Frequency Filters, 11.1.3 Implicit design and generalization in Spreading function representation of operators and Gabor multiplier approximation.

You also have the 2004 paper The convolution theorem for the continuous wavelet transform. My  experience in this domain is limited to localized matched filtering, by  multiplying coefficients by scalar complex numbers to locally modify amplitude and phase (with complex continuous wavelet frames), see Adaptive multiple subtraction with wavelet-based complex unary Wiener filters.

The redundancy of the continuous transform certainly helps a lot. With DWT (a variant of them) I have also used  masks to remove localized high amplitude disturbances in some discrete wavelet subbands. The masks were initially binary ($0$ on the places to remove, $1$ outside), but they were tapered to have a smooth profile between $0$ and $1$. Not very theoretical, but efficient to separate directional data.

Finally, you can find across-scale multiplication for ROI (region-of interest) coding with JPEG 2000.