Oh I know this trope, it's one of my ultimate favorites! It has a name:
https://tvtropes.org/pmwiki/pmwiki.php/Main/BattleCouple
In that link you can find this, but it mainlly western tv, books, comics etc. though and doeant show dramas.
In asian dramas I know only this trope in:
Healer
King of Dramas
....I dont think it's a very popular/often used trope in asian dramas, like it is in western Tv.
Also the turkish drama Kara Para Ask (english name: Black Money Love), is all about it.
The Otp work together and fall in love& at first he's in love with someone else.
Plot: ML & FL both have 1 person very close to them suddenly murdered. They begin to solve the mystery & start to work together. ML is a detective & FL a designer and chaebol's daughter. Features Mafia, betrayals, family and a love that everyone conspires to break apart. The couple goes trough hell and is very much the definition of "MISERY LIVES COMPANY".
....I know you may hesitate to watch because it's a turkish drama, but even of youve never watched turkish drama before, I VERY MUCH RECOMMEND THAT ONE. The drama didnt win freaking A WHOLE 19 AWARDS for nothing!!
It was my 1st Turkish drama ever and it didnt stop me from enoying it (&it was what drew me to also start watching turkis series).
Warning : Like turkish series always are, It's very LONG! (164 eps on Netflix!), but dont let that stop you even if you HATE long series. It's just THAT addicting, no draggy plot at all. It's a real thriller from start to end (well ep3 onwards anyways). The screenwriter is just that amazing.
Major bonus: the OTP's love story buildup (&continuing as a couple) is a slowburn highly believable love story., aka. they got great couple writing. Their great chemistry also doesnt hurt. ;)
If you dont have Netflix, it's also on youtube (there in its original format: 54 eps x 2h episodes
vs Netlix more easily watchable 40 min eps x 164 eps )
Other great recommended "Battle couples" can be found in western tv series Arrow (can also be found on Netflix) (Oliver & Felicity, bonus he's at 1st in love w. another girl in season 1, like you wished, season 2 he starts falling for the new girl)
Plot: Based on DC comics, it's a Bow&arrow weilfing superhero show, where the ML os the superhero & the FL (since season2) is a genius hacker.
and in Salvation (on Netflix too) has a great Battle couple too, and the guy she's 1st in love with is such a jerk that it's VERY SATISFYING seeing the ML steal her away from the jerk ahhahah XD.
Plot: A scifi/political thriller, about an earth-ending asteroid approaching earth, and earth soon to perish and kill all of humanity. the Leads are working to stop the asteroid and fall in love while working together.
Also it's not super long like western tv series can be, it's just 2 seasons.
Also the US tv series Alias, which is about a very cool badass female spy & ML also is a cool (& handsome !) spy, is another Battle Couple. And alao has the trope "stealing him/her away from the he/she is with 1st" like you wished.
...or you can go to that Link I posted above, though I cant guarantee good quality shows from it, I recommend more the ones I mentioned above, which not only meet your criteria but ARE ALSO GREAT QUALITY TV SERIES.
I recommend you to start with Kara Para Ask, it's a can't miss series! ....ESPECIALLY if you love this BATTLE COUPLE trope. ;)
It's such a MUST-WATCH series that I feel so very sorry for those who havent watched it!
