@Stringping. Arggh. So they actually [i]could be [/i](older) Rotos with that dark blue silk? As I mentioned in the first post these mystery strings are clearly nicely low tensioned, whereas the general forum wisdom suggests Rotos are one of the tediously higher tensioned brands. (The only flats I've tried were some neck bending Fenders. No amount of long term, subtle truss rod adjusting has ever returned the JB neck to the perfect state I'd had it in previously )
Either way I think Ernies are the best choice after the much appreciated replies here.
@bigd1 As mentioned again, here and in the orig post, the low tension, neck friendly playability aspect of these strings was important. And as I understand it not all flats are texturally the same (the reason I chose the term 'sheer' in my orig description) and the windings differ - again, why I mentioned their closeness on these particular strings.
Anyway, thanks to everyone for their help. I'll try some Ernies.
Cheers, all.