Try searching the term "Bi-directional Inverter", I think this is what you may be looking for, its a fairly new term to me from a previous term of "True Hybrid" (which doesnt always cover the grid tie with zero export category). I have an SMA inverter that is Grid Tie, it hooks up to the main power wires (from the electric company) in my "fuse box" of the house. I make power and it feeds the loads of my house and when the power of the solar is GREATER than the power my house is using the inverter "throttles back" to no more than the loads. Be aware in order to do this I have had to purchase a $1000 control box and a device that senses which way the power is flowing called an energy meter (around $200) on top of the inverter cost. There is also some associated stuff like internet connections between these devices to make it all work seemless. I can do Zero Export but because clouds cause solar power to "jump" around and there is "lag" time for the devices to throttle back, I choose to do a -4% grid feedback so I am always USING about 200 watts from the grid to keep the power from jumping positive flow (into my house) to negative flow (out of my house). It just helps my application. There are some inverters that claim do exactly this AND also charge and use batteries but I have no experience with them. Solark and the new MPP6048WP (I think), EG4-8KEXP-240 to name a few.
Best of Luck and be sure to enjoy the journey your embarking on, it truely is a "rewarding" journey!
Chuck