Singular noun objects of plural subjects

"they gave their word" is correct, neither because nor despite the fact that the subject is plural, but because "word" is a homonym which in this context presumes the variant meaning "promise" which functions as a logically representative noun (multiple words spoken as a single promise) rather than the variant meaning "word" (a unit of language).

in that logical context only had the speakers given more than one promise each could the author have correctly stated "they gave their words"

by example, of even a singular subject speaking a promise it would be said "he gave his word" despite the fact that we may presume the promise comprised more than a single word

so we see that subject object agreement is not at issue.. rather what is at issue is the essential nature of the variant meaning of the given object "word"

I just read something that caused me to start searching, and I found this thread.

I read, "A survey conducted by the Arizona Superintendents Association of the 2013-2014 school year found that of the 79 districts who responded, 62 percent reported having open teaching positions within their schools, and during the same year, districts and charters reported that 938 open teaching positions were filled by substitute teacher, a 29 percent increase in the number of long-term substitutes from the previous school year." (source: http://www.washingtonpost.com/blogs/answer-sheet/wp/2015/06/19/why-teachers-are-fleeing-arizona-in-droves/)

"filled by substitute teacher" caught my eye as a mistake. But then I thought of the similar-sounding "horse-drawn" (even if pulled by more than one horse) or "stone-ground" (when ground by stones).

You asked What is the rule.

It should be clear from the answers and comments so far that there isn't a single rule: it depends on the circumstances, the intended meaning, on the manner in which the answerer chooses to express it, and possibly on whether they are using British English, American English, or some other variety.

First issue: Collective or individual agreement

You question whether when a group gives its word, it is one collective word that is given rather than each individual's? That would depend on the circumstances. If the group were all together at the time and collectively agreed, it would have been a collective or shared agreement, or collective promise. But if all members of the group were asked individually and all agreed separately, then you have one promise from each person.

Second issue: Plural forms

In this usage, word is not a literal word consisting of individual letters of the alphabet, but rather is a metaphor for promise. Promise has a plural of promises, but word in this abstract meaning does not have a separate plural.

On the other hand, you could not really use promise for collective agreement, because (in this context) promise would be understood as somewhat stronger in meaning than word, and could really only be given individually, not collectively.

Options

So now we can have:

Collective (or shared) agreement: They gave their word (singular for the reasons given above)
Individual agreement:
They each gave their word (singular, because each is singular)
They each gave their promise (ditto)
They each promised
They all gave their word (singular, because there isn't a plural form when used with this meaning)
They all gave their promises (but, personally, I don't think that is a likely expression to be used; and also it is ambiguous because it is not clear whether they each gave one promise or multiple promises)
They all promised (the briefest and most likely expression)

They gave their lives

As Edwin said in a comment "Give one's life is idiomatic" - and that is the standard expression. Idioms do not always follow standard 'rules' and may not strictly be grammatical, and the answer may just be "Because" (i.e. that's just the way it is). Nevertheless, here I can argue that it is grammatical:

Although they may or may not all die together and simultaneously, they each had one life to give, and all of them gave their individual lives. So:

They (all) gave their lives (as for promises above).

Consciousness(es)

First, it's not necessarily appropriate to compare this with the expressions discussed above because they are idioms and this isn't.
Secondly, the Wikipedia reference you mention does not give a definitive answer, and there do appear to be differences of opinion - so what makes you think you can get a definitive answer here?
Brad has already mentioned that abstract nouns do not always take a plural form. With abstract nouns that rarely require or 'need' a plural form, different users may adopt different answers because it is unusual enough for there not to be an accepted 'standard'.

Summary

For a variety of different reasons as discussed, you cannot always project from one particular usage to another because:

idiomatic use may not follow rules;
you need to consider collective v. individual;
there may be no common accepted standard;
there may be difference usages in different parts of the English-speaking world; and
that's the way English is: one of the rules is that it doesn't follow the rules and is not always logical!

In the sentence They gave [object], the object is singular when the group collectively gives a single thing. In your example, They gave their word, the group collectively makes the same promise. In contrast, They gave their lives says that each member of the group devoted or sacrificed their individual lives – even if they shared a common cause.

This is true for all transitive verbs. For example, if a group of comrades make a conference call (or even a series of calls) to a single person, They called their friend. However, if they each make individual calls to separate people, They called their friends.

EDIT: Also note that we don't always pluralize abstract nouns in English. The more abstract the noun, the more likely we are to use it collectively. That's why plural consciousnesses sounds awkward even when talking about multiple minds: We usually think of minds concretely, consciousness in the abstract.

In many cases, you can use a noun either way. Choosing to pluralize or not helps to emphasize whether you mean it concretely or in the abstract. For example, All presidents swear an oath to uphold the Constitution of the United States. This emphasizes that all presidents make an oath of the same nature, even though they all do it individually. They all give their word.

In contrast, we usually write that they devote their lives to upholding the Constitution to emphasize the individual nature of their contributions. You could write that as singular life instead to emphasize the common, abstract nature of their devotion, but we usually don't.