That says the culprit is as I've mentioned, the alternator. Or, I think that it is.
Although you may find that interesting, it kind of makes sense. When you turn the climate control on, the RPM is higher than when it is off, about 100 RPM more. It's like if you'd gently pinch the accelerator pedal. That compensates and makes it not to feel that voltage drop you feel when engine runs on idle RPM, climate off.
And actually, before you get to the alternator itself... how old is it's belt? And are all the pulleys and tightening roller move free? I don't think the problem is with them, but make sure it isn't. It could have been with belt if it happened when you increase the engine load, but it happens when you increase the electrical load. So it's either the alternator, either the battery. What state is the battery in, I don't know if you've mentioned something about it? How old is it, how charged/discharged? Can it be dying, like having a shortcut between two elements inside?