An old A 2002 magazine sports the ad pictured below, juxtaposing
Their hunt the roast vegetable sauce.
with "Our roast vegetable sauce." There's something funny going on in the syntax of the former phrase, that actually made me pay some attention to it. I can't put my finger on what it is though.
Presumably this may be an elliptic construction ("their hunt [after/for/of] the roast vegetable sauce"), where "the roast vegetable sauce" would be in a genitive position. Just as well, this might be just a set phrase, no longer productive.
Is there a name for this type of construction, explaining how this syntax links to a semantics, and/or other examples of the phenomenon?