What is the phrase which means to re-establish a cordial relation between two friends which used to be bitter?
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The term you are looking for is to reconcile:
- When two people are reconciled, they become friendly again after they have argued:
- They were finally reconciled with each other, after not speaking for nearly five years.
My father, a WWII veteran, would have called it "burying the hatchet."
To bury the hatchet means:
to forget about arguments and disagreements with someone and to become friends with them again.
[Cambridge Idioms Dictionary, 2nd ed.]
The phrase you are looking for is "make up"
- to become reconciled (quarreled but later made up) to reconcile with someone; to end a disagreement (with someone).
- to forgive someone and be friendly with them again after an argument or disagreement (from CD)
Bill and Max decided to make up. They made up with each other and are now very good friends.
A colorful way to put it is to "rekindle the relationship".
Oxford definition for "rekindle":
Revive (something lost or lapsed): ‘he tried to rekindle their friendship’
How about reunite or get back together?
reunite: to come together again
Get back together means exactly what the words say.
Or how about bury the hatchet or let bygones be bygones?
bury the hatchet: to forget about arguments and disagreements with someone and to become friends with them again
let bygones be bygones: forgive someone for something he or she did in the past
Another possibility is rapproachement.
rapproachement: establishment of or state of having cordial relations
to improve or repair a relationship that has been damaged by an argument or disagreement.
- She mended fences with her father.
- She and her father are trying to mend their fences.
- After the election, he spent a lot of time mending political fences.
Other related idioms to say are "build bridges", "it is water under the bridge" or "we moved past it", "let's move past it", "let's put it behind us".
pick (something) up where (one) left off: To resume or start (something) again from the last point where one had previously stopped. If it's OK with you, I'd like to go out on a date and try to pick up where we left off! OK, now that the rain's stopped, let's pick this game up where we left off!
After a three-year break in our friendship, Tomás and I picked up where we left off.
To reconnect with old friends
This phrase is more about friends that had a good relationship but lost contact for a while. It's only semi-related to your question since you specifically mentioned a bitter relationship, but it answers the question in the title.