With Apple's changes surrounding iCloud, managing media on iOS devices has become much easier and it is no longer absolutely bound to a single computer. However, I have still found it tricky to sync a single iOS device with multiple computers (different iTunes libraries) without having to entirely wipe the device first.
I synchronize my iTunes library between two computers regularly, so both machines have up-to-date copy of all media files. Furthermore, I set them up with the same username (but different hostname), so the iTunes media file paths are identical between machines.
Searching Apple forums revealed this solution by turingtest2: apparently, each iTunes library is assigned a randomly-generated "library ID". This identifier is stored on the iOS device, so if it doens't match when connecting to a new computer, it will request to wipe the device before proceeding with the sync. Simply copying my entire ~/Music folder from one machine to the other was enough copy all the iTunes metadata (including the library ID) and my device now happily synchronizes on either machine. For those wanting to save time and not copy their entire ~/Music folder, see the forum post for the exact files you'll need to copy.