And, if I saw one walking down the street right towards me, how would I recognise it?

nice stab. what are the most common x-linguistic strategies for the expression of adverbial clauses that deal with time?

Need they involve time-words, e.g. in the sentence "Eating chinchan they swam off", Can I say that "eating chinchan" is an adv that deals with time? After all, it does tell WHEN sth happenes, right?

I don't have the impression that "temporal adverbial clauses" are defined in a technical way. "Adverbial clauses" are clauses embedded in a larger sentence as adjuncts (optional modifiers). "Temporal" refers to time. That's as complete a definition as there is.

I wouldn't call "eating chinchan" a temporal adverbial clause because there is no reason to do so rather than calling it something else. If you have a specific reason to translate it as something like "while", then that would be fine. There's nothing wrong with that label, but it just doesn't seem additionally helpful to me. In that case I'd be inclined to either just call it a "[regular] adverbial clause" or probably call it a manner adverbial or maybe a "circumstantial" adverbial.

In the end, you can look at existing descriptions to see what is described by that term. But there isn't any specific type of construction that absolutely must be called that. It's just for convenience. I would further suggest that most authors who use the phrase do so for a reason because such clauses may differ from, for example, manner, purpose, etc. If you find that they don't differ from those, I don't see the point in discussing it. You might mention that various semantic relationships can be expressed by the general adverbial clause construction in the language and leave it at that. If you find differences, you'll want to discuss them, and in that event you might find the label "temporal" convenient to describe one of the subclasses. It's not more complicated than that.

I think it's a special construction because it involves subordination...

All adverbial clauses involve subordination! (Not all necessarily have a specific marker, but they might.)

chinchan-ACC eat-CONTINUOUS-SUBORDINATION  3PL-PAST-swim distal.deictic
Eating chinchan they swam off

This looks generally like it might be what is called a "converb" or an example of "clause chaining" or "medial verb constructions", or (in Role and Reference Grammar) "cosubordination". The term "conjunctive participle" also appears sometimes (such as for Indic languages), although it's a bit misleading.
It could also be a consecutive marker, but not necessarily if you're calling it "subordination".

In short, these constructions are formally subordination but seem to be similar to coordination in a way (possibly translated as sequential "and" coordination), and they are often translated in the following way:
"Having eaten chinchan, they swam off."
Roughly meaning "They ate chinchan, and they swam off."


Is this the primary way that the language encodes sequence, rather than, for example as in English, with "and" coordination?

Yes, this is the primary way -- tho there is another option, just using the lexical word "after" (without subordination) -- but anyway, that's neither here nor there.

Sounds like possibly two separate sentences with a connecting adverb. "X. Later, Y."
Often that is the source for a coordinator later, but it doesn't sound like this is the main strategy, so I expect that you should consider that 'SUBORDINATOR' to be a primary device for sequences of verbs--

OK, since it involves subordination we can be sure it is an adv clause (i didn't know that adv clauses by def involve subordination! how short-sighted of me!)

Adverbials (of whatever kind) are always embedded, so by definition* they are embedded and therefore subordinate.

(*By the standard definition anyway. This isn't really a fuzzy area, but in general I think the definition of "subordinate" is a little fuzzy in some other ways. Here I'm specifically considering the obvious embedding of any adverbial.)

However, that applies in one direction: all adverbial clauses are embedded. This does NOT mean that all embedded clauses are adverbial. In fact, if this is a "converb" or "medial verb" (etc.) then there are some theories out there that would claim it is not in fact subordinate. That's exactly why the term "cosubordination" was added to Role and Reference Grammar and is very popular in descriptions of Australian, Papua New Guinea and African languages with such constructions.

Interesting idea re converb. I'll look into it. But it's certainly not incorrect to call it an adv clause.

Read about converbs. There may be reasons for thinking of it as something other than an adverbial clause!

and it certain isn't incorrect to say that it has something to do with time.

Doesn't everything? I don't see any fundamental reason to assume it is related to time any more than it would be related to manner or other concepts.

so claim it is an "adverbial clause that deals with time"as Jase suggested doesn't seem so far-fetched, does it?

See above. That's possible, but certainly not something you should assume without working out the details.