Turns out Rylance's speech was actually the prose of a Duluth, Minnesota, poet named Lewis Jenkins.
Rylance chose Jenkins' Back Country to deliver. That short piece follows.
"When you are in town, wearing some kind of uniform is helpful, policeman, priest, etc.. Driving a tank is very impressive, or a car with official lettering on the side. If that isn't to your taste you could join the revolution, wear an armband, carry a homemade flag tied to a broom handle, or a placard bearing an incendiary slogan. At the very least you should wear a suit and carry a briefcase and a cell phone, or wear a team jacket and a baseball cap and carry a cell phone. If you go into the woods, the back country, someplace past all human habitation, it is a good idea to wear orange and carry a gun, or, depending on the season, carry a fishing pole, or a camera with a big lens. Otherwise it might appear that you have no idea what you are doing, that you are merely wandering the earth, no particular reason for being here, no particular place to go."
Rylance said, "I tried one of [Jenkins' works] out at the Drama Desk Awards, and it went down well."