Very Interesting medpb. If it was a see-thru red then it was very likely Rosetta Red. which was a cherry red transluscent and in my opinion one of Egypts very best colours. It could well have had a pair of Armstrong "Egypt Cleartones" pick-up. They usually have no mounting rings and are adjusted for height at the back via a pair of allan screws for each pick-up. Chorr from the Glaswegian band "Battle for Second Place" plays one although his has conventional pickups.
Do you remember where you got it or part-ex'ed it? I dont suppose you have a photo of the bass or remember its serial number because I could probably tell you much more about model designation etc. Without seeing it it sounds like a "Karnak Nefertiti 2" in Rosetta Red if so then it was likely an 1986 model. These were made of Mahogany with a four-piece quater sawn neck, it was a serious piece of kit indeed and as you say a killer shape! Where else are you going to see one! The sound is pretty individual too, my fretless has a strong acoustic responce even before you plug it in. A much woodier natural tone than, by comparison, my Warwick although the Warwick plugged in is the more versatile.
Tis a pity you still dont have your Egypt as they are so rare nowadays, I hardly ever see one. Real collectors oddball, and I have to say I like the oddballs!
Best Wishes
Karnophile