It's an interesting concept, that a tune is identified by the chords one chooses to accompany it with, rather than by its notes.
I had not spotted the relationship between the tune I have sometimes heard called Rolling In the Ryegrass and Green Sleeves, which in the 1580s was apparently a lively jig from the North. Other Scottish children's song tunes which are relatives of Green Sleeves include Green Peas and Mutton Pies, and Who'll Come In To My Wee Ring.
You can choose quite a lot of different chords to accompany them. Some people choose minor chords for Green Sleeves, some choose major chords.
But the tunes still have the same dominant note sequences that show the relationships - they are not the same tune, they are relatives, or variants.