All of the applications that I'm aware of are to other areas of mathematics. Are you teaching number theory? To be honest, if students in a course on number theory require "real world applications", they probably shouldn't be in a course on number theory.
^What do you have against applications?
I first became interested in Diophantine equations in high school when I was assigned two problems for homework. Some might object that they are not true applications.
1)You have forgotten how many eggs you have. You remember the following remainders (% means mod)
What x are possible and what is the smallest possible x?
Some methods of solution lead to a Diophantine equation such as
2)In a sport game one can score a or b points (say 3 and 7)
What scores are possible?
What can you say about the possibilities for low vs high scores?
Diophantine equations are used in chemistry (often not in a systematic way) to balance chemical equations
Pell's equation gives rational approximations to square roots. Which you can contrast with the the Babylonians or Hero's method of divide and average.
The arithmetic application that keeps on giving. The RSA algorithm.
There are many math puzzles that use Diophantine equations. These may or may not interest students. How many ways can so and so... I'm thinking of a number so and so.. Bobs uncle is half as old as his cousin...
this pirate gold question https://www.physicsforums.com/showthread.php?t=85009
If you have not already remind the students that many topics are closely related like
Chinese remainder theorem
Modular Multiplicative inverses
Linear Diophantine equations
I worked with the RSA algorithm. Once I encrypted the entire Declaration of Independence in a single arithmetic RSA operation. I believe I used two 2000-digit primes (from online prime database); the size depending on how large a single message chunk you wish to encrypt. I think it's extraordinary to look at an extremely large, non-random number and consider somewhere in it's detail, lies this important document that is virtually impossible to recover without the modular (Diophantine?) decryption operation, upon which it precipitates out in perfect form.
A number of years ago I posted somewhere on the internet that I do not recall, a treasure map to a priceless treasure but encrypted via RSA and included with it the decryption key. Maybe your students could find it and decode it. Expect a surprise. :)
Edit: Ok, I found it. Jesus, didn't realize how good the search engines are. It's there if they look.