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If you are dating a co-worker, one key thing to do is keep it low-key and quiet. Don't tell people you are dating. There is always the chance you can break up, and you don't want to go through the aftermath that will bring, if everyone is watching and talking about it and taking sides. It's best not to tell anyone you are dating until you are really serious about each other, and make the decision to take things public.
Once you do that, people will start to second-guess the way you work together professionally. That's part of why it's important that you work in separate departments, with far separate spheres of influence.