en_tn_lite_do_not_use/mat/05/40.md

tunic ... cloak

The "tunic" was worn close to the body, like a heavy shirt or a sweater. The "cloak," the more valuable of the two, was worn over the "tunic" for warmth and was also used as a blanket for warmth at night.

let that person also have your cloak

"give that person your cloak, too"