How do appliances prefer to use my solar power from a grid-tie inverter over the power from the grid?
Kirchhoff's current law works for you.
At some point in your installation, the mains supply, the inverter output and the load representing the appliances in your home must all meet. Kirchhoff's law says that the total current going into that point must equal the total current going out.
If your appliances are using more power than the inverter is generating, then it must be true that...
(power from inverter) + (power from grid) = (power to appliances)
If the inverter is generating more than you need, then...
(power from inverter) = (power exported to grid) + (power to appliances)
Either way, as much power as possible is coming from the inverter to the appliances.