Can grid-tie inverters charge UPS batteries if connected to the UPS output?
I thought this kind of function would require additional circuitry to extract the extra incoming power, but from what I've read, it sounds like there are some inverter designs that inherently have this capability.


Yes, As long as the voltage criteria are satisfied a mosfet will happily allow current flow in either direction. So it's not difficult to design an inverter that can feed power in both directions. Indeed APC made some UPSes which they called "delta conversion" that explicitly exploited this, the battery bank was connected to the output through a single converter, then a separate "delta converter" was used to push power from the input side of the UPS to the output side. 

Having said that though, the devil is in the details. I would not connect a grid-tie inverter to the output of a UPS that was not explicitly designed to support it. I would be concerned about the potential for bad interactions between the control systems of the two converters, and about the possibility for excessive voltages on the DC bus of the UPS.