Is it possible to have FBA only some-times?

Question

We are in the process of deploying SharePoint access to remote employees, but are having some problems with the log-in experience.

Currently IIS is set for Integrated windows authentication so that users logged into a domain computer don't have to type their credentials when they open SharePoint.

The problem with that is that when a user logs in through the internet they must type Domain\user to successfully log in and we would like it to be that they only have to type their username.  Unfortunately, it doesn't appear that there is a way around that with just IIS.

So we were wondering if it is possible to have Form Based Authentication (where a user won't have to type Domain\user to log-in) only take affect when users are accessing SharePoint via the internet, and still have Integrated Windows Authentication for the users on the local domain?