- Git Bash
- Git for Windows
- Git Gui
It turns out the solution is to modify a setting on your Windows LiveID account to 'Enable alternate credentials'. You can reach this setting by clicking on your user name in the top left-hand corner once logged into your Live account, selecting 'My Profile', and then selecting the 'Credentials' heading.
Here you will need to click the link to 'Enable alternate credentials' and fill in a password, and secondary username if desired or if the application you use can't use an email address for a username:
This allows the use of basic authentication credentials and fixes the authentication issue with the tools I listed above to manage a Git repository. Make sure to use these credentials when authenticating and you should now be able to manage your TFS-Git repository without authentication issues.