I went to law school in Australia over 10 years ago so things maybe have changed.
At the time we were taught about the Madrid Protocol (US implementation https://www.uspto.gov/ip-policy/international-protection/mad...), which gives you some time to register it in each country after the first registration. If you register it within this alloted time, you retain priority over other people who want to register it too.
You do have to register it, but more importantly, enforce it, yourself worldwide.
IP Lawfirms usually provide this as a service.
On the practical matter of "Is the trademark registered internationally?". Have to do a search in each country's trade mark office. It's very tedious (not every mark is in text, some are pictures, sounds).
At the time we were taught about the Madrid Protocol (US implementation https://www.uspto.gov/ip-policy/international-protection/mad...), which gives you some time to register it in each country after the first registration. If you register it within this alloted time, you retain priority over other people who want to register it too.
You do have to register it, but more importantly, enforce it, yourself worldwide.
IP Lawfirms usually provide this as a service.
On the practical matter of "Is the trademark registered internationally?". Have to do a search in each country's trade mark office. It's very tedious (not every mark is in text, some are pictures, sounds).