What kind of space is u in? Is A a differential operator? Do you know anything about A other than it's a linear operator?
What is the overall context for this problem? Is it from a textbook? If so, how did the book define a linear operator?
In particular, I'm interested in knowing if A always maps real-valued vectors/functions to real-valued vectors/functions. If it does, then you can choose as your real-valued solution. If not, your problem is a bit more complicated.
Note, that, since the real numbers are a subset of the complex numbers, saying there is a complex solution could mean that there is a real number solution. However, the problem says "it also has a real solution" and that is not true.