Huw,

Mea culpa. I've just checked on the 10th and 29th Divisions and you are correct, the 10th (Irish) and 29th Divs were NA & Regular respectively,(I had it in my mind that the 1st RDF and 1st Munsters were in the 10th Div, not the 29th) however I do dispute that the 16th (Irish) and 36th (Ulster) Divisions were TF.
The 16th and 36th Divs were could NOT have been Territorial Force. They were formed from the National Volunteers and UVF respectively neither grouping was part of the Territorial organisation.
A check of the order of battle for each of these divs will show that they were formed of "Service" bns, numbered after the TF bns. I take your point about the "Duration only" men in the 10th, though, so maybe there might have been Nationalist (more likely than "rebel") songs heard on the ship.

Regards

Walrus