I think this is a neat idea. Would be easy to do and could really add a layer of authenticity to what students are doing and working on. Lots of different ways this idea could be used.
"This theory says, "For math to be engaging, it needs to be real. The fake stuff isn't engaging. The real stuff is." This theory argues that the engagingness of the task is directly related to its realness.
This is a limited, incomplete theory of engagement. There are loads of "real" tasks that students find boring. (You can find them in your textbook under the heading "Applications.") There are loads of "fake" tasks that students enjoy."
I agree completely that there are plenty of REAL tasks that aren't engaging, but in my personal experience as a math student and as a teacher that occasionally creates math lessons, I find the most engaging problems are those that have a real application to my personal interests and life. Personally, I believe that if teachers present "real" tasks to the students that they are passionate about and have fun teaching, that rubs off on the students.