Pon tu nombre en alto (2017)

Both series talk about time travel and how it works as a way for ML to become a better person.
Funny "fish out of water" moments and sequences due to the new world that ML is discovering.
Both MLs are sometimes goofy, childish and clumsy, sometimes serious and mature. Both have a somber side too and have a great evolution.
SML has been a friend of FL since childhood.
The main couple have many swoon worthy moments in both series.
Both series talk about health problems of the elderly.

Outro isekai de médicos. Se você conseguiu gostar de Faith, certamente vai adorar Live Up to Your Name.

Both are about Joseon man, traveling to the future, whenever their life in endagered and then meeting there the woman they fall in love with.

the two drama have a completely different cast and concept (the royal concept)
but instead i kept thinking about Live Up To Your Name while watching The King and how the main actors keep going back and forth between the two worlds

Both have a female, which is a doctor. She travels into another world. The man from that world often travels to hers too.

Okay, actually the two shows' story and setting has NOTHING in common... BUT! If you liked this guy in Live Up to Your Name, and you want to see more of him (hell yeah I did!) then I can really recommend this, he was a gem in The Fiery Priest too! <3

both Live Up to Your Name and My Only Love Song are my favorite "time traveling to historical times" dramas.

A man (one from the countryside and one from the past) meets a hard-working woman living in the city. She teaches him the way of the city and ‘civilization’.

And they grow on each other, beautifully.

Romance aside, each drama has its own very unique plot. They have some similarities too.

In both dramas, the FL is a current day doctor and the ML is a doctor from the historical palace who travels to the modern day and tries to adjust to the modern world while falling in love with the FL.

The competition between modern medicine and traditional medicine is seen in both dramas. In korean drama male lead travels to modern era while Thong ek is opposite, female lead travels to ancient era.

Both female leads are doctors. The main couple's relationship and romance is very mature and understanding in both dramas. While Live Up to Your Name is a time-travel romance drama, YAMH is a romance between a female doctor and a special police captain. Both are very outstanding dramas.

Both dramas feature time travel between modern day and a historical time. Both FLs are doctors.
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Both are time travel romances with one half of the couple from the past, the other from present day. Both MLs are connected to the palace but are not royal.

Both are about a Joseon era man who crosses over into modern day Korea and meets the FL.
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Both are time slip dramas with a modern day FL and historical ML. Both dramas have multiple travels between past and present where each of the leads crosses over to the other's time.