There are several instructions on how to connect to TFS Git from Xamarin Studio if you're using the Team Foundation Service.

We have locally installed TFS 2013 which also has Git support, but Xamarin Studio is unable to connect to it with the following error: Version control operation failed: authentication not supported.

Is there a way to work it out?

You need to enable HTTP basic authentication in TFS

解决方案：打开TFS所在服务器的IIS，展开到tfs的虚拟目录，右侧功能视图，身份验证，将“基本身份验证”启用，xmarin studio 就不会报“authentication not supported”的错误了，会弹出登陆密码输入框。