For some odd reason it seems that the system designed into TortoiseGit doesn't allow the user to interact with git when it requires user interaction. For instance, accepting self-signed certificated is not possible, which gives you the known issue 318.
As of TortoiseGit 126.96.36.199, it is still not possible to accept certificates through the GUI. But it is possible to get your git repository to work with TortoiseGit (and work with the required certificate).
You will need to have msysgit installed and available in your PATH for the following to work.
The first step is to run some git command, for instance
git clone https://myserver.com/depot via a command line so that git may auto-accept (or ask you to accept) the certificate. This step is crucial to get the certificate details saved onto your machine.
What you will want to do next is go to
C:\Program Files (x86)\Git\.subversion and copy everything into
%USERPROFILE%\.subversion. Basically, this should copy the certicates that were accepted by msysgit so they can be used by TortoiseGit.
Another, and possibly better solution, is to create a symbolic link so that those 2 folders are in fact a single one. For instance, you could do something such as
move %USERPROFILE%\.subversion %USERPROFILE%\.subversion_backup mklink /D %USERPROFILE%\.subversion "C:\Program Files (x86)\Git\.subversion"
which will make
%USERPROFILE%\.subversion point to your
C:\Program Files (x86)\Git\.subversion folder. This has the benefit that any future certificate will work both for msysgit and TortoiseGit.
Thanks to Mexx’ C# Corner for pointing out the solution.