I have an outside light that is controlled by a light switch inside. I took the light out and put a adapter to plug in an extension cord to run LED Christmas lights. When I turn the switch off, just the first strand of lights glows very faintly so I wonder is my switch used as the ground and the lights are finding ground at the tree they are hung at.
If the switch is a dimmer, remote operated switch, or a switch with a neon backlight, it may allow a small amount of leakage power even when off. It's not enough power to light an incandescent light, but it may be enough to dimly light led lights.
Try running an extension cord over to a standard outlet and troubleshoot your lights in that configuration before you bring the adapter and switch into the picture.
You can also separately test the switch and adapter by plugging in a standard incandescent shop light.
The key is to isolate and remove as many variables as possible from the problem.
I have the same situation. When I turn my Christmas lights off outside by an outdoor multi plug switch they glow very faintly. I noticed that some LED strands when unplugged from the wall altogether still glow faintly for a minute. So I don't think its a leakage as much as power still in the power cords, probably 100ft or more when you add it all together and it just takes a while to dissipate. Even if it was a leakage it's so minute it would be negligible. Maybe .01 of a watt. If you had a multimeter you might be able to see it, maybe not.