There are several MSS settings of tunes for the air Callino Casturame from the late sixteenth/ early seventeenth century called Callino, Calleno, or Callinoe. These are cited by F W Sternfield in Elizabethan and Jacobean Studies OUP, 1959 as
a) CALLINO ROBYNSON for cittern, in Cambridge University Library Dd.4.23
b) CALLENO for lute, in the William Ballet Lute Book in Trinity College Dublin. D.1.21
c) CALLINOE for lute, incomplete MS in Cambridge University Library, Dd.3.18
d) William Byrd's CALLINO CASTURAME for Virginals in the Fitzwilliam Virginal Book at Cambridge UL
e) & (f) two unidentified MSS of Byrd's setting
g) 'An Irish Tune CALLINO CASTORE ME' in Playford's Catch that catch can, or the Musical Companion published 1667
Sternfield notes that tunes (a)'Robinson', (b) 'Ballet' and (d) Byrd are variants of the same tune ( a sound-file played by Donald Sankey of Byrd's setting can be found here:- Byrd's Keyboard Music), while the Playford tune is different altogether.
It is important to note that these are all English settings of a tune which may have Irish origins but which an expert in early Irish music commented as having "no feature which might be accounted distinctively Irish" ( Aloys Fleischmann, Sources of Irish Traditional Music c.1600-1805. Garland, New York 1998).