RequireJS and legacy application

I have a legacy application and I have refactored parts of the application into separate backbone.marionette applications. I do not have the time or the budget to refactor the whole thing and I want my code to be easier to manage which made me think of requirejs.

Most of the files are minified and munged together.

Can I use requirejs for this type of hybrid solution where I can work on separate backbone modules and still access the existing javascript?

As someone who just recently started using Require.js on a legacy, Backbone-using codebase, I feel your pain :-) I used a combination of approaches, which I'll list here.

Let's say you have fileA.js and fileB.js, and you want to convert fileB.js to use Require, without changing fileA.js:

Abuse the global space

Require doesn't force you to import every variable through it; even in a Require-ified file, you can still access global variables the same way you would with non-Require-ified code. This means that if fileA creates all of its variables in the global/window namespace (which is very likely if you weren't using Require before), fileB can access them whether or not fileA uses Require.

This wound up being my solution for most of my legacy files; I just left them as is, and put all the new Require-ified stuff below them. That way every global they create is ready and waiting when the Require-ified files need them.

Now, this is great if fileB depends on fileA, but what if it's the reverse? Well, Require also doesn't prevent you from making new global variables, which means that fileB can share anything it wants with fileA, as long as it is willing to put it in the global space.
Duplicate code

Don't get upset; I know how important "DRY" coding practices are. However, for just a few files what I wound up doing was making Require-ified duplicates. This wound up being necessary because I'm using a Handlebars plug-in for Require to do my template compiling, so if I wanted any file to use Handlebars I needed it to be Require-ified.

To combat the normal un-DRY issues, I added comments to the old files effectively saying "don't add anything to this file, the Require-ified version is the 'real' version". My plan is to slowly convert more of the site to Require over time, until I can finally eliminate the original, obsolete file. We have a small shop, so it works for us, but at a larger company this might not fly.
Refactoring

I know, you said you wanted to avoid this, but sometimes a little refactoring can give you a lot of bang for your buck. I personally barely refactored anything at all, but there were just a couple places were a small tweak greatly simplified matters.

Overall I see refactoring as something you do after you switch to Require (to slowly over time bring your non-Require-ified code "in to the fold").
Shims

Chchrist is correct in saying that shims are a good way to solve the "half-way to Require" issues However, I personally didn't use them at all, so I can't really say much about them except "look in to them, they'll probably be helpful".