I think one could ameliorate that somewhat with the most direct possible lighting and the most matte finish possible on the background to minimize reflectivity. One should be able to refine the area affected by the operation using Saturation as a criteria, and since anything reflected onto the coin would be not nearly as saturated as the background itself. Alternatively, one might select a background color of a pure RGB hue, and desaturate that color only in postprocessing enough to eliminate it from the coin while leaving enough of it on the background to select. Shadows from the coin will interfere either way, though. That could be helped by raising the coin physically from the background with something like a well nut, as I do: That helps to create a physical demarcation between coin and background, and although I've never used it to this end it might be relevant. This is an interesting intellectual problem only for me, since I've no Ancients around to experiment with, and enough trouble keeping track of what I already like without adding yet another category. For the record, I use the Gimp myself.
For those who want to play, I used a feature of my camera I have never used before. These were shot in camera reducing to this size so I might post them here. CT will not accept my camera's full size. I always shoot large and reduce after trimming so it will be harder to trim cleanly here. 1 and 2 are shot on a black background but not shaded completely so they may not be 000000 level black 3&4 are laying directly of a textured gray felt. 5&6 are directly on a piece of wood. 7&8 are on a white light box but show the problem of flare from all that white reducing the contrast of the image (ruining it IMO). 9&10 have the intensity of the white background light reduced and shows glare back from the ring light but that can be post processed out. With care, I probably could get rid of that flare but I'm not into white enough to try right now. As soon as I upload these, I will finish off the black ones just to see how long it takes. I am not playing with the gray and wood. They are trash to me.
Better with less contrast? I usually have to tweak such things a couple times to get what I want but rushing is never a good idea. I am always happy to discuss these things on private conversations but there is no reason to do that if you are bothered by experimenting and just want something good enough to use on eBay. I admire many of the photos posted here by several of our members (TIF and AJ come to mind even though both use that reflection trick whch I dislike).
Plain backgrounds are the way to go. I also find its much easier to fill the background then crop out the image. Sometimes I have to zoom in to get the small pixels on the edge with the wand. A busy background makes it nearly impossible to crop.
I just came across this thread. Very interesting. Here's a video I did of what I would do to separate the coin from the background. I don't have an ancient coin to shoot. I chose this half dollar because of the reeded edge and how you can retain it. A perfect circular crop would either force you to lose the reeded edge or you would have some of the original background color showing, without a lot of cleanup work later. This video shows a mostly automatic way of removing the coin no matter what shape it is from the background. I used a Samsung Tablet with the screen set to white using a free app I downloaded for it. Here are some of the rubber stoppers I use to separate the coin from the background.
I love RR coins but I always dread "picture day" with a new serrate denarius for this exact reason. They always take at least twice as long as regular denarii to get the background cleaned up.