This sounds like mostly
an issue for the WinINET and SChannel teams at Microsoft; getting a packet capture trace of the scenario (without Fiddler involved) may help clarify what specifically is happening on the client side.
you put the client certificate in Fiddler's folder, does everything work without any problems at all, or do you find that periodically authentication starts failing completely? If authentication always works in this case, that points to one type of problem; if it periodically fails in the Fiddler configuration, that points to a different type of problem.
By default, Fiddler doesn't support prompting for client certificates via the UI. However, if you'd like, you can install an extension that does this.
1. Download http://fiddlerbook.com/dl/ClientCertPicker.zip
2. In Window Explorer, right-click the file, choose Properties. Click the "Unblock" button at the bottom of the window.
3. Open the ZIP file and extract ClientCertPicker.dll to the %USERPROFILE%\Documents\Fiddler2\Scripts folder.
4. Restart Fiddler. Fiddler will now prompt for a certificate.
If you're interested in the source code for this extension, please see http://fiddler.wikidot.com/certpicker
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