A backwards relationship
Haskell, 120 111 bytes

import Data.List
a&amp;b=(a#b,b#a)
[]#_=[]
(a:y)#b=[a]%y where p%(i:w)|reverse(i:p)`isInfixOf`b=(i:p)%w;p%x=p++x#b


Test runs:

λ: "abc bab"&amp;"abdabc"
("cba bba","badcba")

λ: "abcde"&amp;"abcd bcde"
("dcbae","dcba edcb")

λ: "hello test"&amp;"test banana"
("hello tset","tset banana")

λ: "birds flying high"&amp;"whistling high nerds"
("bisdr flyhgih gni","wihstlhgih gni nesdr")