Did it fully charge that battery, i.e. did it stay on for long after it was charged in strip A? Are you sure you're getting full power out of strip B? Did you measure it with a voltmeter?
Plugging in an unpowered charger into a device like a cell phone can give some squirrely results, at least based on what happens when the power fails in my house when the phone, laptop and a bunch of other gear is being charged.
One last question - are the plug blades on the charger polarized? I can't remember what allegedly unpolarized device I had that would not work correctly if it was plugged in upside-down (relatively speaking).
Other than that, I'm officially out of ideas. Sorry I couldn't help. I bought a USB charge monitor (and lots of adapters) so I could monitor the current flow into the device being charged. Helpful in situations like this and only cost $6. To diagnose this problem correctly I'd use those along with a DVM to measure the voltage at strip B. But remote diagnostics on a text-based system where you can't even draw a sketch easily - yes, I give up. (-:
I would recommend getting another power strip and seeing if it exhibits the same characteristics as the one you have now.