Quit Calendar and also the Reminders application, if it's running. Force quit if necessary.
Back up all data.
In the iCloud preference pane, uncheck the boxes marked Calendars and Reminders. You'll be prompted to confirm that you want to delete your iCloud calendars and reminders from the computer. They'll still be on iCloud.
Relaunch Calendar and select
Calendar ▹ Preferences ▹ Accounts
Click the plus-sign button at the bottom of the account list on the left (which may now be empty) to add a new account. Select iCloud and enter your credentials. Your iCloud calendars should appear after a brief delay.
Do the same with Reminders.
Credit for this solution to ASC member melisolaris and others.