SEOUL—South Korea announced on Friday that people visiting restaurants and cinemas and other public spaces will have to show vaccine passes, amid a surge in COVID-19 infections and six confirmed cases of the Omicron variant. The government also re-imposed limits on private gatherings, which had been recently relaxed, as the country posted record numbers of new cases this week. Desperate to fend off the Omicron variant, authorities halted quarantine exemptions on Friday for fully vaccinated inbound travelers and made a 10-day quarantine mandatory. From next Monday, people visiting 14 designated public spaces, including hospitality and entertainment venues, will have to show their vaccine passes, Prime Minister Kim Boo-kyum told a COVID-19 response meeting, setting out the plan to reduce the risk of community spread. The public will have a grace period of a week to get used to the new rules. While people have been required to show their vaccine …