Just a question, since I am not a dual citizen and have no experience with this. When traveling abroad (excluding the US), do you actually need to show the foreign country's border control the same passport that you've given to the air carrier in order to check-in/board the flight?

I realize that in US pre-clearance at Canadian airports the CBP officer does verify that the passport information associated with the boarding pass matches the passport being presented for entry into the US. But in many foreign airports I have not been asked for my BP. Off hand I can only recall UK and Japan asking, Finland as well but only to show an onward connection outside of the Schengen zone. Many countries don't have any form of APIS, so why would e.g. Argentina border control have an issue with you presenting a British passport for entry when a Canadian passport was shown to AC?

Conversely, can't you change your passport info prior to checking in for the return flight to avoid the need for an ETA if you had a British passport on the outbound?