Is there a family of English-speaking nations, united by cultural values, liberal market economies, the common law and democracy, that is waiting to welcome Britain with open arms after it frees itself from the constraints of the European Union? Some optimists think so.

David Davis, the British government’s Brexit secretary until he left his post this week, said in a 2016 speech on the referendum: “This is an opportunity to renew our strong relationships with Commonwealth and Anglosphere countries. These parts of the world are growing faster than Europe. We share history, culture and language. We have family ties. We even share similar legal systems. The usual barriers to trade are largely absent.”

In 2013, Boris Johnson, a prominent Brexit supporter and foreign secretary until he also resigned this week, described Britain’s joining the precursor to the European Union in 1973 as when “we betrayed our relationships with Commonwealth countries such as Australia and New Zealand.”

Both departed in protest of Prime Minister Theresa May’s most recent plan for Brexit, which includes concessions to European Union rules to make trading with the bloc easier post-Brexit and which, in Mr. Johnson’s view, has “suffocated” the dream of a global Britain, trading with the wider world.