Update...
The customer interested in the bike returned just as I started the third replacement; B*gg*r, now I was under pressure; unlike the tyre*.
New tube in tyre, after checking both tyre and rim for anything sharp and finding nothing; pump up; nothing. Put tyre on bike and start on front tyre.
Three minutes later rear inner tube explodes: there's a loud "pop" and dust comes out of the rim.
Tyre off, held carefully in the same position, checked, also in same position. 1 cm long split in inner tube, on the inside. check wheel. I can see exactly where it was because there's a mini explosion of dust. Nothing sharp or untoward.
Find new tyres from store, and new inner tubes, and dig up some fresh rim tape. Fit everything onto bike.
That was midday. As of 16:00 the tyre was still solid. we'll see what happens tomorrow. If it stays up I'll go with
@Dogtrousers theory: the other tyres were haunted.
Oh, and the customer decided the bike was too big...
*Sorry...