Apocalypse(2009)

Apocalypse

Influenced by two of their friends, a pair of female high school students, Natsu and Kumiko, begin playing the popular MMORPG "Apocalypse Online." When they log in for the first time, they find they've been given a rare item, a pair of "marriage rings," which gives the two of them special bonuses when they are near each other. But while their "relationship" is a funny quirk of the game to Natsu, Kumiko secretly longs for it to become a reality...

(Taken from Mangaupdates)