To have OsX share the tmux clipboard, you need to run a script that syncs both buffers.
You can create a login item for you user that runs run_tmux_copy.
Contents of run_tmux_copy and tmux_copy are below.
Make sure to place them something useful, like /usr/local/bin.
The reasons for all of this are documented in the repo for the reattach-to-user-namespace command