I think that this is new ground without a consensus yet. For example, Wikipedia doesn't mention this on its GUI Widgets list.
You've got the ASP.NET "Autocomplete Textbox". I think that name works, but is not completely clear as to whether the textbox accepts multiple values or not.
If you allow new values to be entered, "autosuggest textbox" may be good too.
Other examples include entering e-mail addresses in most modern applications (Outlook, Gmail, etc.). I worked on a project where our key feature was such a pair of autocomplete textboxes, but I can't remember if that's what we called them.
Perhaps if you want to be clear that you are autocompleting multiple values you should call it a multiselect autocomplete textbox. That is wordy, but at least one other person has called it that on the Internet already.