Subsite has "own" menu, but not different primary menu?

I'm looking for a way to present different primary menu items, or at least secondary menu items based on path, so for example on example.com/chicago I show the menu structure:

-Neighborhoods
--Southside
--Lakefront

but on this path: example.com/nyc I show this structure:

-Neighborhoods
--Chelsea
--Midtown

Is this possible in Subsites? Seems like a common use case yet I can't find a simple way to do it which might be because Drupal caches menus. Possible at the theme layer?

I'm using a Superfish dropdown menu included with the Acquia Marina theme so the child items are drop down leafs from the primary items.

I know this can be done using Domain Access but I think that's really overkill for the job.