I looked high and low for some advice on how to make this work, but couldn't find anything obvious. Then I stumbled across an obscure reference to setting up sharing with Anonymous domains in the Office 365 Admin Center.
- Login to the O365 Admin Center.
- Navigate to the Exchange Admin page.
- Go to Organization settings, then Sharing.
- Edit the default sharing policy under individual sharing.
- Add a new sharing rule.
- Select "Sharing with a specific domain", and enter "Anonymous" as the domain.
- Save everything, then go back and share with a non-365 user.