This is an excellent question Segun. First of all, your typical inverter in the market has the capacity to charge your batteries through the AC input. So you can actually charge your batteries via solar and the grid or NEPA.
Before I answer your question, just know that the system without any input from the NEPA is called a completely offgrid system.
Now can we charge with both Grid and solar at the same time?
Yes you can!
If you do this, then you may not have to by the correct number of solar panels needed for your load. NEPA will augment and charge your batteries faster.
Any implications?
There would have been implication if your inverter and charge controller is not smart but this is most times not the case. So your inverter senses when the battery is fully charged and it stops once it gets to 100%. The same applies to the solar charge controller. The controller will automatically detect when the battery is fully charged and it will cut off charging.
 
So even when you charge with both NEPA and solar at the same time, the inverter and controller will stop charging once the battery is fully charged.

How about the battery? Can it handle it?