A breakthrough Friday in grueling divorce talks between the U.K. and the European Union opened the door to the next stage: tough trade negotiations that will determine Britain’s economic relations with Europe, the U.S. and the rest of the world.

After six months of tense negotiations, the EU and the U.K. overcame acrimony to make progress on a handful of contentious issues around Britain’s planned departure from the bloc in March 2019, including a financial settlement and the rights of British and EU citizens to live and work...