Most residential solar PV systems are simply grid tied, and residential grid tied inverters cannot do that; the amount of power they produce is governed by the amount of insolation on the array, though SolarEdge inverters can with the addition of some hardware. Many inverters can throttle back output in response to frequency munging, but one obviously cannot change the frequency as long as the inverter is connected to the grid.
The adjustments are to the battery management systems, not to the inverters. Most residential grid tied inverters have nothing to adjust.
The only one I know of is SolarEdge, and the inverter alone cannot do it; you must install their energy meter.
Perhaps I'm the odd one out, but I think you should stop saying these things, or think about how to say them more accurately. What you say is true for the majority of legacy inverters, but it is and will be increasingly untrue for newer ones. As you acknowledge, Solaredge can follow a metered load, with the addition of the meter accessory, the last point being a rather minor detail in the context of this discussion. The Enphase IQ8, hitting the streets this month, will also be able to do it with an appropriate meter setup (Envoy and CTs). (I believe IQ7 already has some ability to do it, but I haven't used those grid profiles.) Those two manufacturers represent about 90% of the US residential market share, so it is more or less now a fact that most new residential solar inverters
can 'do that' [i.e. in Wayne's words "use a pair of CTs to measure the house load at each point in time, and ... provide current that matches the house load (up to the inverter's maximum rating)"]. Also, the comment about frequency munging leaves out the point that interactive inverters can be connected to non-grid primary sources, e.g. a battery inverter in an off-grid or temporarily islanded setup, which is also an increasingly common setup I'm sure our OP will want to eventually understand.
The statement "The adjustments are to the battery management systems, not to the [solar] inverters" is, well, only correct
usually. In the case of SolarEdge, there is only one inverter for both, and the behavior (with meter installed) depends entirely on the programming. If you want a zero export or limited export system, you can set that up, and this has been true for more than 4 years, with or without battery. This will soon be true for new Enphase systems as well. Just because all of this isn't done that often (yet) does not mean that solar inverters can't do it. It's never been impossible in theory, and increasingly it will be possible with most off the shelf products.