I've setup a SharePoint 2013 farm that has two web applications, one for the main site collections for the business (intranet), and another that holds the user profile service (intranetpersonal). Both web applications are using claims based authentication and in an attempt to rectify this issue I've already setup the intranetpersonal web application to allow anonymous access to all sites.
Everything works fine in Internet Explorer (feels weird saying that in a web context, heh) however whenever I view a newsfeed, staff directory, or anything that accesses the user profile service Chrome isn't loading the images. I've checked the console log for Chrome and it looks like the following error is the cause:
Resource interpreted as Image but transferred with MIME type text/css: "http://intranet/_layouts/15/1033/styles/Themable/corev15.css?rev=BdxJNFd%2FTPOed3Z8IKEJ9A%3D%3D". init.js?rev=zwpf9CD1m7am6imImmQglQ%3D%3D:1 GET http://intranetpersonal/User%20Photos/Profile%20Pictures/testinguser_MThumb.jpg
Is there a way to rectify this issue, or am I up the creek without a paddle having to only support Internet Explorer within the organization (shudder).
It turns out this is a Google Chrome bug which is already addressed (it's resolved in the Canary build so it should make it to production in the near future). For future visitors I would advise looking at Robert Lindgren's answer as that is the typical resolutuion to this kind of an issue.