I took 2 real analysis classes in undergrad and I thought it was lots of fun... but no it's not very helpful in physics. The concepts and theorems from those courses haven't applied to anything I've encountered in physics. Basic definitions like "inner product" might show up in analysis and physics but the uses are different. I've been wrong many times before so don't take my post as an official answer, but if look in any "math methods for physics" book, none of the topics will come from analysis... it's all DEQ's, linear algebra, special functions, vector calc etc. (perhaps "advanced analysis" or graduate level stuff subsumes DEQ theory and L.A. etc, but undergrad analysis is mainly about what the real number system is and developing Calc 1-3 more rigorously).
But complex variables are used a lot in graduate physics (contour integrals, residues)... so the complex analysis class I had was helpful. However, real is probably a prereq for complex and those math methods books would cover all the complex you need to know for physics anyway.