Despite the names of these fields, as a student I found real analysis more 
abstract than abstract algebra: *real* analysis was less real and more abstract to me than *abstract* algebra. I don't think I can justify this, but let me give two examples:

 - [Lagrange's
   theorem](https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory))
   in abstract algebra: The order of a subgroup $H$ of a finite group
   $G$ divides the order of $G$. Sure, this is abstract, but it is
   discrete and definite and understandable from a thorough grasp of cosets.
 - [Heine-Borel theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem)
   in real analysis:
   Closed and bounded iff every open cover has a finite subcover.
   Requires understanding limit points, accumulation points, triangle
   inequality.

Certainly one can pluck out a theorem from abstract algebra that is decidedly 
more abstract than a particular theorem in real analysis, to make
the opposite point. But to me
abstract algebra as a whole was (and still is) more concrete than real analysis.

So I would argue: Abstract algebra before real analysis, just because 
proof sophistication would improve in abstract algebra to help with the
more difficult (and abstract) proofs in real analysis.