You could keep your two identities separate, but if you're looking for a job, your reputation may actually help you!
You can clearly see I use my real name on the Stack Exchange network of sites. When I first started, I used a online handle I've been using for years. However, as I got closer and closer to thinking about looking for a new position, I figured that this was the perfect demonstration to prove to future employeers that I could troubleshoot, critically think, work in a teamed environment, learn from my own mistakes, etc. I decided to change my name to my real one.
When I began looking for jobs, I assumed that employeers were going to be Googling me as I applied. They would be stupid not to. However, it never even dawned on me to include anything about Stack Overflow on my resume.
I started interviewing with a company based in Tampa, FL, who was interested enough to fly me down for an interview. One of the first questions during the interview was, "So, tell me about your time on Stack Overflow?" I was a little surprised, but it ended up with the company offering me a job. I have no doubt that my Stack Overflow profile helped with that decision on their part.
TL;DR Your work on Stack Overflow can be helpful in your job search process. Don't count it out and don't be embarrassed about the question you ask. Just your active presence here shows to employeers that you are consistently learning and evolving.