Despite that, I have on occasion trained both immediately after eating, and on an empty stomach, and I didn't notice any impact on performance. Not that it's optimal for muscle gain, but sometimes life means you're short of time and can't do everything optimally, and it's better than doing nothing. In the case of training on an empty stomach, that's where it's important to eat something immediately afterwards.
As for benefits of training fasted, there aren't really any. You can find some studies saying it increases metabolism of body fat in that moment, which is true for endurance based exercise, but only total calories in vs out over the course of the day, week, or year affect total body fat gain/loss anyway, so it really doesn't matter even for that.