Although the sentence is weird, I am wondering if it's grammatical:
The branch was cut with a sword wielded by a swordsman.
The problem is that "wielded by a swordsman" complements an indirect object "a sword", not sure if this is allowed. I am thinking it is, but we can never be too sure.
"wielded by a swordsman" is a participle phrase acting as an adjective on sword.
The branch was cut with a sword. The sword was wielded by a swordsman. Thus:
The branch was cut with a sword wielded by a swordsman.
You can parse it as
(The branch) (was cut) (with a (sword (wielded by a swordsman)))
It's an odd phrasing, but it's perfectly grammatical.
The branch was cut with [a sword wielded by a swordsman].
It's perfectly grammatical, but a tad odd semantically (see Andrew's answer).
But "sword" is not an indirect object (there is no indirect object anywhere). Its function is that of complement of the preposition "with".
The bracketed element is a noun phrase in which "sword" is modified by the past-participial clause in bold. Semantically, past-participials (and gerund-participials) as modifiers in NP structure are similar to relative clauses, compare a sword that was wielded by a swordsman. And past-participials are bare passives, as your example shows.
Agreed, nothing wrong with it. It's just redundant, as a swordsman, by definition, cuts things with his sword. However, since it's a bad idea to go around hacking up trees with one's sword (as it's not made for that task) it might not be clear from swordsman what tool was used.
So it's semantically odd -- suppose he used an axe? Then why call him a "swordsman" at all?
Anyway, the real awkwardness comes from the passive voice. If you change it to the active voice, you can write the sentence to include an explanation:
Since no other tool was available, the swordsman used his sword to chop down a few branches to make a fire, silently cursing the hours he's have to spend sharpening it afterward.
