Leadfingers, accompaniment does not necessarily mean chord symbols. This song sounds like early 19th century, in an idiom which in my opinion lends itself to harmonisation, though I agree it does not need it. There are other songs, not necessarily much older, where "functional harmony" is completely off.

Consider the following little choir arrangement that I just wrote; is it a possible interpretation, considering that we have a romantic ballad rather than a sea shanty?

X:1
T:Constant Lovers
I:abc-version 2.1
I:abc-creator MidiZyx2abc 1.07
I:linebreak $
M:3/4
C:Trad. England, Arr: Grishka, donated to the public domain
L:1/8
Q:1/4=110
%%score [ Soprano Alto Tenor Bass ]
K:Bb
V:Soprano
%%MIDI program 52
"^Very freely"F2| B2d2c2| B2A2B2| G2G2F2| E4E D | C2D2E2| F2G2A2| B2c2A2| % 8
w:As I was a-wal-king down by the sea-shore, Where the winds and the waves and the bil-lows did
B4A B | c2F2F2| c4c d | e2d2c2| d4f d | B4B A | G4G G | % 15
w:roar, There I heard a strange voice make a ter-ri-ble sound, Like the wind and the waves that did
(cd) e2c2| (B2A2)G F | (B3cd2| F6)| G2F3E | D4F F | G2c2B2| A4F2| B3c d2| % 24
w:e--cho a-round,_ Cry-ing: "Oh___ my love is gone! He's the youth I a-dore, He's gone and I
E e3d2| (dc) B2A2| B4|] % 27
w:ne-ver shall see_ him no more!"
V:Alto
%%MIDI program 52
F2| D2F2E2| D2F2F2| E2C2D2| E4C =B, | C2C2C2| C2F2F2| F2G2C2| % 8
w:As I was a-wal-king down by the sea-shore, Where the winds and the waves and the bil-lows did
B,4C C | C4C D | E4E F | G2F2E2| F4A2-| A A G2F2-| % 14
w:roar, There I heard a strange voice make a ter-ri-ble sound, Like_ the wind and
F F E2D B, | G2G2C2| (E2C2)E C | D4z2| (D3EF2)| E2D2C2-| C2B,2B, B, | E2E2G2-| % 22
w:_ the waves that did e-cho a-round,_ Cry-ing: "Oh, oh,__ my love is_ gone! He's the youth I a-
G2F2A,2| F3E F2| C G3B2| (=Bc) F2E2| D4|] % 27
w:-dore, He's gone and I ne-ver shall see_ him no more!"
V:Tenor clef=treble-8
%%MIDI program 52
F2| F2G2A2| B2c2d2| B2G2=B2| (c2e2)G G | c2c2c2| e2d2c2| f2_f2e2| % 8
w:As I was a-wal-king down by the sea-shore,_ Where the winds and the waves and the bil-lows did
d4F G | A4A B | F4c B | A2B2c2-| c2B2A F | d4f d | B4B d | % 15
w:roar, There I heard a strange voice make a ter-ri-ble_ sound, Like the wind and the waves that did
(c=B) c2g2| (_g2f2)c A | F4z2| (B3cd2)| G2A2A2-| A2G2F F | B2G2e2-| e2d2c2| B3A B2| % 24
w:e--cho a-round,_ Cry-ing: "Oh, oh,__ my love is_ gone! He's the youth I a--dore, He's gone and I
G c3d2| g2d2c2| B4|] % 27
w:ne-ver shall see him no more!"
V:Bass clef=bass
%%MIDI program 52
F,2| B,,2B,,2C,2| D,2C,2B,,2| E,2E,2F,2| (G,2C2)C, C, | C2B,2A,2| G,2F,2E,2| D,2C,2F,2| % 8
w:As I was a-wal-king down by the sea-shore,_ Where the winds and the waves and the bil-lows did
B,,4F, F, | F,4C, B,, | A,,4A,, G,, | F,,2G,,2A,,2| B,,4D, F, | G,4D, D, | E,4G, F, | % 15
w:roar, There I heard a strange voice make a ter-ri-ble sound, Like the wind and the waves that did
(E,D,) C,2E,2| (F,2F,,2)G,, A,, | B,,4z2| (D,3F,B,2)| E,2F,2^F,2| G,4D, D, | E,2C,2E,2| F,4E,2| % 23
w:e--cho a-round,_ Cry-ing: "Oh, oh,__ my love is gone! He's the youth I a-dore, He's
D,3C, B,,2| E, C,3G,2| E,2F,2F,,2| B,,4|] % 27
w:gone and I ne-ver shall see him no more!"