Trademarks are territorial, and an EU trademark protects you only within the EU - the same goes for any other trademark from any other jurisdiction.
For this reason, there's no such thing as a "global" trademark. However, it is possible to own multiple trademarks in multiple jurisdictions. The choice then depends on the coverage the business has as well as the budget, as each additional trademark will be priced separately.
If you require trademark protection in countries or regions outside the EU, you have to file separate trademark applications in those specific jurisdictions. You can do so either by filing individual applications with the local intellectual property offices or by using international trademark systems such as the Madrid System to apply for multiple countries at once.