Handle file download from ajax post

I have a javascript app that sends ajax POST requests to a certain URL. Response might be a JSON string or it might be a file (as an attachment). I can easily detect Content-Type and Content-Disposition in my ajax call, but once I detect that the response contains a file, how do I offer the client to download it? I've read a number of similar threads here but none of them provide the answer I'm looking for.

Please, please, please do not post answers suggesting that I shouldn't use ajax for this or that I should redirect the browser, because none of this is an option. Using a plain HTML form is also not an option. What I do need is to show a download dialog to the client. Can this be done and how?