You do not. iPhones are designed to sync to only one library. Attempts to sync to another library will wipe all the content from the device and replace it with content from the new library.
If you have content that you want on the device, add it to your computer and iTunes library. This assumes that you have a legal right to the content. If you did not buy the content, you DO NOT have a legal right to it.
Apple does not prevent users from putting content on there devices that the user has a legal right to.
Per your own statements, you do not own the content and therefore have no legal right to it.
If you want the content, purchase it yourself and then sync it to your device.
First, this is a year old thread, so I'm uncertain what was intended or expected by adding a silly, uninformed comment to this old thread.
Second, syncing and manually managing are not the same thing.
Nothing in this thread mentions manually managing content.
Finally, it has already been clearly stated how to sync content from a new computer. To be blunt, copy the ENTIRE iTunes folder from the old computer to the new computer.
The iphone 5 has essentially merged syncing and manual managing. With the old ipods, you could plug it into new computers and manually add/remove content easily. You can't do that with an iPhone without syncing and erasing, why Apple would do that I cannot for the life of me understand, unless it's about forcing users to purchase from the iTunes store.
And, to be blunt, copying the entire iTunes folder is hardly an acceptable work around. What if I have a large library and I don't copy files to the iTunes media folder?
The iphone 5 has essentially merged syncing and manual managing.
No such thing has occurred.
If the user wishes to manually manage content on the device, they simply select the option on the Summary tab to Manually manage content on this device. The caveat is, they cannot "sync" with any computer, they must manually manage content on any computer they decide to connect to.
copying the entire iTunes folder is hardly an acceptable work around. What if I have a large library and I don't copy files to the iTunes media folder?
It is a perfectly acceptable way to use the device and iTunes, frankly that's the way it is designed to work.
Consolidate the library and allow iTunes to move all the media in iTunes into the iTunes Media folder.