Anyone carry a folding tire?

My work commute is usually 20 miles per day, and I typically have two
panniers, one my laptop/office bag. On the ride home last night I hit a
piece of glass that not only punctured the tube but ripped the tire
open, so in addition to my tube sticking out of the cut the tire's torn
threads were also poking out.

This is my first flat in probably 15 years (!), and was beyond repair as
I couldn't fix the torn tire. I'm sure some duct tape or something
similar applied in the tire would have let me limp home on a low PSI
tire, but I didn't have those materials handy.

My question to all you commuters: do you carry a folding tire for cases
like this? They look a little pricier than normal tires, and I'm
wondering if this is overkill on my part or if that's something regular
commuters carry in a pannier.