Eli Zaretskii wrote:
> I'm building Emacs with GnuTLS 3.3.11 since many moons ago.

OK, I installed the attached patches to GNU Emacs master. The first one requires 
GnuTLS 2.12.2 or later. The second one uses the GNUTLS_NONBLOCK flag, and works 
around the apparent GnuTLS bug with EINPROGRESS and ENOTCONN.