This question arose while posting today's word a moment ago.

An award given by a council is "the council's award'.
An award given by several journals is "the journals' award'.

But what of the Phoenix Award given by the Council of Learned Journals?

Is it the Council of Learned Journal's Phoenix Award?

Is it the Council of Learned Journals' Phoenix Award?

The second option.

"The Council of Learned Journals' Phoenix Award" is correct because the award is given by the Council of Learned Journals (plural). Otherwise it would be the award given by "the Council of Learned Journal".

In cases like this it is worth recasting it, at least in your head. It would then become "The Award of the Council of Learned Journals". The first try would become "The Award of the Council of Learned Journal", and is obviously wrong.

Arnie: the award is given by the Council of Learned Journals (plural).

But arnie, the Council is singular entity, not a plural one, and it is the council that gives the award.

Confirming this, note that you would say, "The Council of Learned Journals gives an annual award. This year the Council's award goes to ..."

Your construction makes it seem like the journals, not the Council, are giving the award.

If the name of the entity is "The Council of Learned Journals" then the possessive (which is unlikely ever to be used in the USA) would be "The Council of Learned Journals's award"

Clumsy it might be but as the possessive of a singular compound noun ending in "s" that's what I fear it should be. As it would be were it to be the Jones's award. Because it's hard to imagine any situation where the plural noun "journals" can be a singular possessive the construction just looks wrong.

Frankly I'd re-write the sentence.