Hey Dennis. What you're seeing under the bike is the condensation of the vapors from the fuel tank. The purge valve / canister system on the V-Twin RT might have looked good on paper, but in real life it was about as useful as a soup sandwich. One thing is that it only taps into, and uses, vacuum from the rear cylinder. So almost every chronic problem - rich or lean - with the rear cylinder can be traced back to that. And there is no check valve in the vent line from the canister to atmosphere. So, when the purge valve opens it not only sucks in vapors and any liquid from the canister and fuel tank, but also fresh air. Some of us, maybe a lot of us, have just removed the system. Do a search for Canisterectomy. I'm not saying that your problem is solely the purge valve system. Just that removing it eliminates that piece of it.
It's a long story that covers everything from gas smell, to boiling in the fuel tank, to fires and a recall in 2013. But essentially, underbody temps and fuel tank temps are the root of all evil for the bike. There's such an excessive amout of evaporation from the fuel tank that the canister is useless. If you get down at floor level after a ride, and look at the butt end of that rubber hose that sticks out from below, you'll see the dripping coming from there. Contrary to a once popular believe, it is NOT due to overfilling the tank. It's just condensation of fuel tank vapors.
I can't tell for sure by looking at your picture if you have the air scoops from the 2013 recall installed. They fit previous year RT's, also. They go into the body work under your mirrors and direct airflow under the body. They're not just vent holes but actual scoops. They sit where the original fixed air deflectors are located and are black unless painted. They help a lot with all of those heat issues. Do a search for information for that. You'll be glad you installed those if they aren't already there.
You might want to try some Berryman's B12 if the Seafoam doesn't yield results soon. And maybe consider replacing the fuel filter. There's a lot of garage time on that, too. Right side, right behind the canister. And followup with those vacuum hoses. The 2012 is the best of the V-Twin RTs. You just need to get this little glitch taken care of. If you do your own maintenance on that beautiful bird behind you, you can fix a Spyder. We had a 75 Cardinal for over 20 years. Decided to see the world at ground level for awhile.