I had a go at harmonising this a few years ago using an anglo concertina without ever coming to a satisfactory result so I forgot it and sang it unaccompanied which actually is fine.

Going through this thread I thought I might try again. My earlier results were unsuccessful for two reasons, I think. First I made the mistake of thinking "one sharp, it's in G (doesn't end on E so not Em). Then I realised it ends in D and it's not it's a mixolydian mode tune with a D final. So I tried on that basis and peppered it with D, G and Am, but the Am just did not seem right - like Phil said at the beginning, it's not that it doesn't harmonise with the melody, it just doesn't feel right.

In fact, mixolydian is a major mode so you need major chords and these fall on I, IV and VII rather than the I, IV and V of a major key. The other thing I have found from some reading is that with modal tunes where the chord on the 5th degree is minor, the equivalent to a major V7-I progression is VII-I and I've tried it regularly in other tunes and it works.

So based on that I came up with this:

[D]The [Dsus4] gal[D]lant frigate, Amphi[C]trite, she lay in [D] Plymouth Sound,
Blue Peter at the foremast head for [G] she was [C] out[G]ward [D] bound;
We were waiting there for orders to [G] send us [C] far [G] from [D] home;
Our [Dsus4] ord[D]ers came for Ri[C]o, and thence a[D]round Cape Horn.

I've used Grishka's Dsus4, though my first thought at that point was G and it will work.

I'll try and put a recording on soundcloud sometime but I'm off to see Martin Carthy at the Welly just now (weather permitting)