De côté.
After reinstalling Git and TortoiseGit several times, trying different repositories and looking for ideas on other bug reports, I managed to fix this by editing the shell icon overlay identifiers on the Windows registry.

What I did to fix it was:
1 - Open regedit;
2 - Navigate to HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\ShellIconOverlayIdentifiers;
3 - Rename all entries that start with "Tortoise" so they start with "01_Tortoise";
4 - Kill explorer.exe/tgitcache, restart explorer.