Recently after extending a web application that uses Managed Metadata Navigation and enabling anonymous access on the extended site, the global navigation for anonymous users on the extended site disappeared but was still showing on the original web application.
The problem was two fold, firstly when creating a term in the navigation term set using the Term Store Management tool, SharePoint by default uses the Term-Driven Page with Friendly URL setting which will only be visible to users of the site that have write permission, so pretty useless for an anonymous site and secondly any dynamic/dropdown menus are not displayed after the level 1 term.
To fix the first issue is relatively straight forward, just use the Simple Link or Header for the term and browse/provide the URL to the target page when the user clicks it. So this fixed/enabled all the level 1 navigation to re-appear but the dropdown menu items (level2, level3 etc) were still not being displayed.
The second issue was a bit more troublesome but after numerous different approaches I found the quickest fix (and the only for the extended web application site) was to edit the master page of the original site, change the SharePoint:AspMenu element within the master page and set the UseSimpleRendering to false for this element.
Save, check-in and publish the edited master page and then after applying some CSS mark up to get the global navigation dropdowns to match your desired look and voila the navigation works on the extended site!