If you have a portfolio denominated in, say, GBP, and you invest in a CAD-denominated bond, then hedging into the portfolio’s base currency would be hedging the CAD/GBP exchange rate so that your returns in GBP are the same as your returns in CAD.
Hedging the basis currency, simply means reducing the exposure to certain key risk factor which is a foreign currency in this situation.
You may take direct hedging by shorting the long primarily exposure in foreign currency (e.g by entering into Forward contract) or take a cross hedge by shorting another currency which is closely correlated to that particular foreign currency.
You might reasonably decide to not hedge at all and just wait that everything to revert to the mean.
another point is, and what i dislike about this method of conveying words is ive always been used to looking at it as Domestic/Foreign. Keeps it simple but I understand in certain situations it can change.
What’s irritating is that in one Level III reading it uses the terms correctly, while in another reading it uses them interchangeably (but has a footnote that says that they’re different).