It's not a "real" projection in the sense that it is mathematically stretching a sphere onto a plane. It looks like someone was inspired by Goode's Homolosine projection and did something vaguely like that, but without an understanding as to WHY Goode split the map the way he did.

One possible way to try to get things back together (assuming that you don't have local-area maps) is to print the map above, fold it so that the split areas line up, then take a picture of the area of interest. There will be some distortion, but it's a realtively low-tech way to get the source info back.

The other way, of course, is just to slice and dice in an image editing program.