In case it will help someone else, the problem turned out to be in the FastCGI auth server's printf, the last "statement" of the HTTP header should end with \n\n instead of \r\n.
The following was wrong:
printf("Content-type: text/html\n\n"
"Set-Cookie: name=AuthCookie\r\n"
"<html><head><title>FastCGI 9010: Hello!</title></head>\n"
...);
This did the trick:
printf("Content-type: text/html\r\n"
"Set-Cookie: name=AuthCookie\n\n"
"<html><head><title>FastCGI 9010: Hello!</title></head>\n"
...);
Thank you!