Most likely the wizard who cast Imperio would tear their soul.
The closest situation to this is the Dark Lord using the basilisk to kill Myrtle. It was most likely her death he used to create a Horcrux out of his diary, which would prove it did split his soul. The basilisk is of course not human and therefore may be unable to get a damaged soul, but it would show that using something or someone else to kill would split the soul of the wizard ultimately responsible for the killing.
When being controlled by a Parselmouth, the basilisk simply follows the commands of the person directing it.
“He knew what was happening, he could sense it, could almost see the giant serpent uncoiling itself from Slytherin’s mouth. Then he heard Riddle’s hissing voice: ‘Kill him.’
The basilisk was moving towards Harry, he could hear its heavy body slithering ponderously across the dusty floor. Eyes still tightly shut, Harry began to run blindly sideways, his hands outstretched, feeling his way. Riddle was laughing …”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)
Similarly, someone under the Imperius Curse is simply carrying out the will of the person who cast it, with no control over their actions. It’s, in that way, analogous to how the Dark Lord used the basilisk to kill Myrtle.
Therefore, from the situations we’ve seen, it’s likely that the wizard who used the Imperius Curse to turn someone else into their tool would spilt their soul (and likely be able to make a Horcrux from the kill as well), while the Imperiused wizard’s soul would remain intact.