Ah, got it. Because of the vowel beginning ils, you say the final t in veulent (but not the s in ils; I don't know why) - so it's like 'veul-t il une chemise'. But when you slur it together at speed it sounds like 'veul tilune chemise'.
To me, that t sounds like the ts in 'tsunami', and I think it's fair enough if they don't make a clear and precise 't' sound. After all, in English I can say 'they wanna'.