This seem to be a regression on macOS Sierra. A solution that worked for me has been mentioned by Josh McGinnis https://github.com/tmux/tmux/issues/543:
brew install reattach-to-user-namespace
Ensure the following is set in .tmux.conf:
set -g default-shell $SHELL
set -g default-command "reattach-to-user-namespace -l ${SHELL}"
In .vimrc or ~/.config/nvim/init.vim (for Neovim):
set clipboard=unnamed
Now all is well and I can copy/paste between system <-> vim sessions using vim keybindings and/or system ctrl+c / ctrl+p.