I’m currently TRE on IAA licence and I can sign EASA and UK licences (at least until end 2022), so if someone had both licences and it was all in date and appropriate and my airline allowed it (and it will) then I would sign both. Because it would entail extra paperwork (two notifications, post check reports etc) I would expect extra beer.
Once I get my UK licence back I will put my uk number on the uk signing and my IAA number on the EASA signing.