@benjistokman Kirchhoff always holds! The issue in the real world is what happens when you don't have
dedicated circuits or really understand where your loads are. What matters is assuring that no segment of wiring significantly exceeds its rating. Under ordinary conditions, this is what the breaker assures, since all power reaching the wiring passes through it from the utility. However, once you have a source of power that's inside from the breaker, most common breakers will be oblivious to it. So, let's say you have a 15 amp circuit at 117 volts. That's a max of 1755 watts. Now you turn on that inside air conditioner that is pulling around 1200 watts. Let's assume your plug in solar is running super great with no losses and providing the full 1200 to that circuit. So far so good, you should be pretty much power the air conditioner from the solar. The breaker is seeing practically nothing from that circuit via the utility. Now, realistically, you probably have other stuff on that same circuit other than the air conditioner. Your wiring (let's assume it's in great shape and connectors are all clean and tight and so forth) now has a margin of a bit over 500 watts. As you now use other appliances or whatever on that same circuit, you're pulling power from the utility being watching by the breaker. Let's assume you use more than 500 watts now from other stuff (actually it's going to be a significantly, maybe much lower limit, due to real world losses, old wiring, etc.). Every watt over that 500 (coming from the utility) is pushing you over the rating of that 14 gauge wiring, which is getting hotter with every watt. So let's say you have 700 watts of other stuff turned on too. You're now 200 watts over the maximum rating of the wire, but the breaker doesn't know that, because it sees a much lower current passing through from the utility. Does an extra 200 watts matter? Depends on the condition of wiring, etc., but some equipment pulls much more power than we normally think about, especially as starting surges. Anyway, the bottom line is that this kind of setup creates a situation where specific segments of wiring can be carrying far more current than their maximum ratings, and the associated breaker cannot be expected to trip. As I mentioned in the OP, there are devices that try to deal with this, some involving adding equipment to the panel, and at least one (rather expensive) that seems to use a different technique. But ordinary consumers likely won't know any of this stuff. They'll usually just open the box pull out the panels and the inverter and plug it in.

@lauren oh you are talking about 2400 watts consumption and 1200 watts production on the same rail. Not 1200 each. That's a problem yeah.