As a physics student I've had a one-semester course "Mathematical Methods in Physics" where one third of the course was about complex analysis. Basically, this is about integration in the complex plane and how it can help you to solve integrals on the real line which are too hard to solve without the tricks of complex analysis. It has other applications too, for example easily solving Laplace's equation in 2D systems (or 3D systems where the solution is symmetric in one direction).
You might not understand what I'm exactly talking about, but let me state that complex analysis should belong to the backpack of any mathematically/theoretically inclined physicist.
It is true that for this course real analysis was no prerequisite, but by far most of the students had taken the class (probably simply because it was offered sooner in the curriculum than this course, quite logically). Having followed the course, I can say that you can indeed get by without real analysis, the reason for this being explained later in this post.
That being said, I also took a complex variables course in the mathematics department (also a one-semester course, but this time of course not simply a third of it). For this you definitely needed a real analysis course.
Why the difference? Well, actually, both courses talked about the same things, the latter just in far more depth. The first didn't pay attention to any rigour, it simply wanted to get the mathematical theorems across and how you could use them. The second course was a course for math students, i.e. everything you used you had to prove. Personally, I preferred the latter, as the extra time and attention for rigour showed the interconnection between the different theorems and at the end gave a deep feeling of elegancy: complex analysis is probably the most elegant course I've taken so far. In the "mathematical methods in physics" course I often felt like a robot applying mysterious tricks, not knowing why they worked (well okay, I did as I had taken the mathematical course before this one, but all the rest felt like a robot then).
That being said, I can't decide based on the name of your course which of the two types it is, but now at least you know what the two possibilities are. I hope that helped.
PS: sometimes I've called it "complex analysis", sometimes "complex variables"; they're synonyms!
PPS: I haven't answered your question "which to take: PDE and complex variables, or something like real analysis" and I find it very hard to answer! PDE is without a doubt the most useful one, but a course like real analysis makes you more mature mathematically speaking, which is also very useful, but maybe in a less direct way. I've already described the importance of complex variables. If you have the time for all three, I'd do all three! (and put real analysis first, in that case) But maybe more precise advise can be given if you describe where your interests lie, where you want to go (maybe rigour is, say, repulsive to you).
