If a company is listed on a stock exchange in its home country at a price of say 1.50 Euro but is listed on NYSE at a price of $20. How does this work?
The price varies a lot, so as an investor why would I go to the NYSE price when I can have it for 1.50 Euro?
The shares can be the same in all respects. If I remember correctly, BHP is dual listed in Oz and the UK, and the shares are exactly the same - but the prices can be significantly different: a few points apart, which is a difference worth having.
I can't remember what the story is with dividends though. Surely franking (or the lack of it) would be a big complication?
To avoid a really stupid mistake, I would buy all my stocks on one board.
If that be NYSE, so be it.
I would learn the tax laws and stuff relevant to that board.