Better information as well as temporarily triggering ssl_verify should be possible. See docs for webkit1[1] and webkit2[2].

[1]https://developer.gnome.org/libsoup/stable/SoupMessage.html#soup-message-get-https-status
[2]http://webkitgtk.org/reference/webkit2gtk/stable/WebKitWebView.html#webkit-web-view-get-tls-info