Turn off power strip when another circuit is turned off
Sure, this makes perfect sense. A DAYTON 5X847 relay would do the job. Its coil rating is 10VA at 120VAC, so you don't need to supply DC to the coil. When it receives 120VAC, it switches its DPDT contacts.

You would wire up the DPDT contacts to the live and neutral of your secondary circuit, routing them either nowhere (off) or to the power strip.

The relay is rated at >20 amperes so it will handle a full residential circuit.