I think the other replies have covered the fact that unpermitted grid tie is just too risky to impossible. Smart meters will detect it immediately, and analog meters might run backwards undetected until they send someone out to read it.
So that leaves the solution of having a second independent grid at home. You'd still need the inverter to output alternating current, and then you'd need breaker boxes, wiring, and outlets, so lots of extra costs. But theoretically, this works. For example you could wire up your storage shed, and then plug in a freezer and fridge set really cold. When the sun shines, they run and hopefully keep cold until the next day with sun. The one thing that solar does well is handle reduced load: if you don't have something drawing an excess of solar power, it doesn't hurt the panels or the inverter.
Someone mentioned the problem of power cycling and its effect on appliances, and I think that might be an issue as well. Appliances just aren't designed for intermittent power. I think with the spread of solar, that will change, we will get "smart" appliances that turn on when power is available, even optimize their running to use such power, and tolerate when it isn't.
But without batteries, such a system isn't really good for much. It's not like you see the sun shining, so you go use the second microwave that's plugged into the solar electricity when you want to make lunch, or worse need to unplug and move the appliance. I guess you could wire the solar system next to certain appliances, then you just have to plug and replug them in. For example, you know the sun is shining, so you go to the laundry room, replug the washer into the solar power and do laundry.
But really, solar panels only make financial sense when you can use all the power they produce. So without batteries, you'd need a load that soaks up all the solar power but doesn't mind the intermittent production. That's why I suggested a freezer, to "store" the energy as cold. An electric car might work too, but then you're limited to charging at home when it's sunny. An electric car is just a battery, but it does let you convert that stored energy to mobility.
As mentioned, the other use for solar electricity is air conditioning. Again, it doesn't work to either have 2 air conditioners or to flip a big switch every time you want to use solar, it's just not practical. However, they do make a mini-split unit that takes direct current from solar panels and alternating current from the grid. It uses the solar power when available and complements it with grid power when not. I have one and it works, it runs fully off of 3 300W panels in series (so 900W at around 30-35V DC), and even when it runs off of the grid, it is very efficient. But even that is an inefficient use of the solar panels, so I also store the power in batteries (through a charge controller) and I run 12V DC lights in my house from it. This is a network of 12V LED light fixtures, with their own wiring that I ran through the attic and their own switches. So that uses the solar power to offset some grid usage in my house, but with the cost of batteries and charger, it's still not a cost savings, it's more for a grid alternative backup during extended outages.