their relationship after the incident were or was?

Simply, "was" is used for singular objects, and “were” is used for plural objects. In your sentence fragment 'their' > pronoun; 'relationship' is the object was/were refers to and is singular.

Their relationship after the incident was...

Additionally, 'Was' is used in the first person singular (I) and the third person singular (he, she, it).

'Were' is used in the second person singular and plural (you, your, yours) and first and third person plural (we, they).