On our first mission, we encountered a, hmm, what would the word translate to, a Projector perhaps? Not like the tool, but one who projects. The primary creature is actually quite large, perhaps twice the height of a human, nearly ten feet tall with long arms, a tan fur across the whole body, and an elongated snout-like face with small sharp teeth. They have the power to create illusions, mental landscapes of various sorts. They're not particularly complex, the creatures primarily use them as a hunting tool, overlaying details on the landscape that camouflage them or make them entirely invisible while the victim may not even realize that they're seeing an illusion (unlike the more complex fair that one gets from, say, Asterion). They normally hunt on the other side of the galaxy and exist in a sort of half-physical, half-etherial state, their primary prey being various kinds of pyschic parasites they can devour. 

The ab-Enkiji had some kind of small device that was attuned to the realms of one of these creatures and they activated it when we broke into their lab. The Projector was enraged at first. It projected an illusion as it entered of itself suddenly disappearing to hide its attacks and we had to fend it off. It eventually calmed itself after smashing most of a lab and hurling away several guards that we directed it towards. When it did, we were able to communicate with it in basic terms and it let us activate the Enkiji device again so that it could escape. 

Althoug the Enkiji had activated it themselves, it was much easier for them to explain to their superiors that the Phantom Thieves had summoned some sort of monster that had wrecked their lab and that they had no responsibility for the damage whatsoever, which only added to our reputation.