Hmm...are you talking about the operating system's tabs, or Vim's tabbing system? Creating a new tab (in something like Terminal) and running vim won't share registers, because you'll be running two separate vim processes. Running two tabs within the same copy of vim should share without problems.
To put that another way, type:
:tabnew
...into vim. Is the result familiar? Do you get the same kind of tab you're describing?
If not, that's the problem. Type :help tabpage
to learn about vim's tabbing system, which will share registers & more.
(If so - if that is the same kind of tab you're talking about - then the mystery continues...)