Traditional grammar leads us to analyse a structure in completely different ways according to one small thing such as what part of speech it is introduced with. However this structure is interesting because what is essentially the same structure is found in different languages, but introduced in different ways.
In Latin we have:
The ablatives of a participle and a noun (or pronoun) are used to form a substitute
for a subordinate clause defining the circumstances or situation in which the action
of the main verb occurs. The ablatives are only loosely connected grammatically to
the remainder of the sentence, hence its name absolute (absolūtus = free or
unconnected).
An Ablative Absolute with a perfect passive participle is widely used in classical
Latin to express the cause or time of an action:
Hīs verbīs dictīs, Caesar discēdit. With these word having been said, Caesar departs.
Acceptīs litterīs, Caesar discēdit. With the letter having been received, Caesar departs.
Leōne vīsō, fēminae discessērunt. With the lion having been seen, the women departed.
Equally common is an Ablative Absolute with a present active participle:
Leōne adveniente, fēmina discēssit. With the lion approaching, the woman left. "Ablative Absolute" in Latin Library
In each of these examples the verbal form is definitely a participle as it agrees with the noun. Note that although we do not have ablatives, with+noun phrase is a valid translation. In Ancient Greek they do not have ablatives, but they use a genitive instead.
In Scots Gaelic the same structure is found, but introduced by agus "and". The semantic relationship between the main clause and the absolute clause is a bit variable as in the English structure. Here it is used to describe circumstances. Most commonly it is used to give a consequence:
Thachair seo air Là na Bliadhn' Ùire ann an 1919 agus seòladairean a' tilleadh bhon Chiad Chogadh. This happened on New Year's Day 1919, and sailors at returning from the First War. BBC News Website 5/9/2018
A' tilleadh translates literally as "at returning" (so tilleadh is technically a gerund) but usually corresponds to the use of the present participle in English.
Bha iad a' tilleadh. They were returning.
Just as in English it can be difficult to tell the difference between a gerund and a present participle.
Thus we have the same structure introduced in different ways:
- Ablative + noun + participle in Latin
- Genitive + noun + participle in Ancient Greek
- "And" + noun + "at" + gerund in Gaelic (where "at" + gerund is the equivalent of present participle in English.
This shows that looking at the introductory word is not necessarily the best way to analyse the structure.