The Cambridge Grammar of the English Language (Page 1266-67) has this section:
10 Verbless clauses
We confine our attention here to verbless clauses in dependent or supplement functions comparable to those realised by non-finite clauses, as described in the main part of the chapter.
(a) Complement to with and without
 i They were standing against the wall [with their hands above their heads].
ii They were wandering around [without any clothes on].
iii [With the children so sick,] we weren’t able to get much work done.
iv Who is that guy [with his hands in his pockets]?
The [boldfaced] clauses have subject + predicate structure, but with no verb in the predicate. With and without do not license finite complements, but non-finites are found in addition to the verbless forms (see §8.3 above). The bracketed PPs function as adjunct to a clause ([i–iii]) or post-modifier in NP structure ([iv]). With is semantically similar to have, and without to not have: [i–ii], for example, entail They had their hands above their heads, They didn’t have any clothes on.
CGEL is saying the boldfaced portions are verbless clauses and "have subject + predicate structure". But is that true?
As suggested by CGEL, with/without can be replaced with having/not having in [i, ii, iv] as follows:
(1) They were standing against the wall [having their hands above their heads].
(2) They were wandering around [not having any clothes on].
(4) Who is that guy [having his hands in his pockets]?
Here, can you say that having take verbless clauses as complements? I don't think you can. In (1), (2), and (4), I think that having takes their hands, any clothes, and his hands, respectively, as its direct objects.
Note, on the other hand, that with cannot be replaced with having in [iii]:
(3) ??[Having the children so sick,] we weren’t able to get much work done.
In [iii], unlike in [i], [ii], or [iv], the children so sick is a verbless clause and does have subject + predicate structure.
But is it correct to treat the boldfaced portions in [i], [ii], and [iv] as verbless clauses as well?