terminal command to open new chromium tab

A couple of options to consider...

1) Find an existing instance of Chromium, activate the window, and send Ctrl+t to open a new tab:

xdotool search --onlyvisible --name 'Chromium' windowactivate --sync key --clearmodifiers --window 0 ctrl+t

2) Open the about:blank page in a new tab:

chromium-browser 'about:blank'

#1 does exactly what you want.
#2 is cheesy.

Open a New Tab and look at the Developer Tools - Sources tab. URL for New Tab is:

chrome-search://local-ntp/local-ntp.html

This will open a new tab in existing Chromium instance:

chromium chrome-search://local-ntp/local-ntp.html