SoapUI not working with fiddler for REST service testing

I just tried to do this myself (using Fiddler and the request library from npm). Here's how I got it working:

process.env['NODE_TLS_REJECT_UNAUTHORIZED'] = '0'; // Ignore 'UNABLE_TO_VERIFY_LEAF_SIGNATURE' authorization error

// Issue the request
request(
{
    method: "GET",
    uri: "https://secure.somewebsite.com/",
    proxy: "http://127.0.0.1:8888" // Note the fully-qualified path to Fiddler proxy. No "https" is required, even for https connections to outside.
},
function(err, response, body) {
    console.log("done");
});

This is with Fiddler2 using the default port and proxy options (and no proxy authentication).

What is the site's URL?

It is probably caused by either of these two issues: http://blogs.msdn.com/b/ieinternals/archive/2009/12/08/aes-is-not-a-valid-cipher-for-sslv3.aspx or http://blogs.msdn.com/b/fiddler/archive/2012/03/29/https-request-hangs-.net-application-connection-on-tls-server-name-indicator-warning.aspx

The old workaround is to configure Fiddler to only use SSL3 when talking to the host in question. The newer workaround is to either use Fiddler4 with the latest .NET4.5.2 framework, or if you're using Fiddler 2.5.1, see the "SNI Hack" section of http://www.telerik.com/blogs/what-s-new-in-fiddler-4-5-1

In your OnBeforeRequest event handler, add the following code to fix the issue for certain sites:

if (oSession.HTTPMethodIs("CONNECT") && oSession.HostnameIs("BuggySite.com")) 
{ 
  oSession["https-DropSNIAlerts"] = "yup"; 
  FiddlerApplication.Log.LogString("Legacy compat applied for request to BuggySite.com"); 
}