Shooting Stars & Comets have tails.
Here is Example Wiki Image showing Comet with 2 tails :
With the Earth revolving around the Sun , is there some [invisible] "tail" like Comets have ?
It might be invisible to earthlings. It might be outside of visible spectrum.
It might be visible in heat or infrared or ultraviolet Image techniques , when visualized or recorded from Moon or Mars ?
A different take to the dust tail answer - Earth has a very well known "invisible tail": the magnetotail.
Earth's internal magnetic field compresses ahead of the Earth in its travel around the Sun due to interaction with the solar wind, and is drawn out into a long tail behind.
This kind of structure is not unique to Earth.
(1) Strangely , Earth has a Dust tail though it is not similar to the Comet type !
A Comet will have tails made of Matter coming out of the Comet itself.
Earth has a tail which not made of Dusty Matter coming out of the Earth , rather it is Matter "floating" in the Solar System which is getting swirled or rearranged by the Earth.
Earth tail is not visible to naked eye :
"Earth has a dust tail not because the planet itself is particularly dusty, but rather because the whole solar system is. Interplanetary space is littered with dusty fragments of comets and colliding asteroids. When Earth orbits through this dusty environment, a tail forms in the rear, akin to swaths of fallen leaves swirling up behind a streetsweeper."
https://phys.org/news/2010-11-earth-tail-alien-planets.html
(2) Stranger Cases are Moon & Mercury which have the Comet type of tails !
Moon tail is not visible to naked eye :
https://www.wral.com/story/mercury-s-tail/20260519/
Mercury tail is somewhat visible to naked eye :
