I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly correlated with some measure of size, and things can be improved by using intensive variables, e.g. dividing everything through by some measure of size. E.g. if your units are countries, you might divide by population, area, or GNP, depending on context.

Oh - and to answer the second part of the original question: I can't think of any situation when adding the product of all the correlated regressors would be a good idea. How would it help? What would it mean?

I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly correlated with some measure of size, and things can be improved by using intensive variables, i.e. dividing everything through by some measure of size. E.g. if your units are countries, you might divide by population, area, or GNP, depending on context.

Oh - and to answer the second part of the original question: I can't think of any situation when adding the product of all the correlated regressors would be a good idea. How would it help? What would it mean?