I believe that your question can be stated "Can we detect a lingering effect on the local environment in interstellar space a decade or two after the passage of a relativistic starship?"
The first step is to ask just how relativistic? If it's just "really fast" (say, 0.99c) there would be one set if effects, but if it was much fast -- 0.99999c, for instance -- some new effects would come into play.
The second is to ask about the density of gas in the vicinity. If it's like normal interstellar space (typically a million atoms per cubic meter) its passage will be much harder to detect than if we're in something like a Giant Molecular Cloud where densities can be 100 to a million times greater.
First off, you won't see any lingering electromagnetic radiation: That will necessarily be light years off by a decade later and you specified local effects. That leaves two things that I can see: (a) Effects on the pre-existing gas, and (b) bits of ship that remain behind.
The interstellar medium is nearly entirely atomic hydrogen, either neutral or ionized. The temperature is typically 100-1000 K. The mean free path of these atoms is very large: 105 miles up to a million times that. The mean speed of the atoms is about 1 kps.
That means that any effect the passage of the ship has on the local interstellar medium will be entirely erased by the diffusion of the interstellar medium (IM) in a short time. Imagine that the ship blows a tube of vacuum in the IM a million miles across. Neutral hydrogen atoms from outside of this region (just a few mean free paths away) will move in at 1 kps. In a few million seconds, the void created by the ship's passage will be refilled. (A million seconds is about 10 days, so a single year is plenty of time for the IM to diffuse back to equilibrium.)
If the IM is much denser, the effect may linger as long as a decade, but we're not talking about the IM anywhere near us.
The second potential source is bits of the ship left behind. Hitting the IM at .99c is precisely the same as the IM hitting the ship at .99c, and that's quite hard radiation -- the IM's protons are hitting at 5 GeV. Unless the ship has force shields of some sort, this is likely to cause some spallation -- atoms of the ship's front end eroded into space. This might be detectable, but the problem I see is that those bits are unlikely to be dumped into space with zero transverse velocity and will thus quickly leave the local area and be undetectable a decade later.
The ship it may dump enough energy into the IM to heat it -- creating a long tube of hotter than normal IM along its path. The problem -- again -- is that the mean free path is so long that the hot atoms would immediately decamp for distant places and the colder atoms outside the ships's range would diffuse in, erasing the tracks.
If the speed were insanely high (as if .99c wasn't insane enough), possibly you might see some sort of effect where a bow wave of plasma builds up big enough to clear a really wide tube along the path, long enough that it can't fill in in a decade, or enough energy is dumped into the IM that it's still excited a decade later. The problem is that that energy -- which is large -- is coming from the ship's kinetic energy through friction with the IM. (a) This slows the ship and (b) it probably does very bad things to the ship's structure.
The last effect is that we might see some side effect of the ship's drive or shields. But since they're basically magic (Clarke's Third Law) I don't see how anyone can give you a scientific answer.