If you have non-UV flash tubes (let's put aside the issue of fluorescent tubes, this just confused the issue!), emitted UV from the flashtube will make any cloth with UV brighteners fluoresce as described in RDKirk's post. If you send that UV light to a subject with brightened clothes, or to textiles for a catalog, the flourescence of the subject will render inaccurate all color rendition in the fluorescing object. If you send the UV light to a light box made of material with brighteners, it will fluoresce and throw off the balance of the entire subject, not just causing the clothing/textile to fluoresce, since it is the illuminating source (which is off balance by its own fluorescing!).
Since I never have used any flashtube without UV filtration, I don't know if you bounce a non-filtered tube against a softbox with non-brightened material, will the resulting light at the subject still have any UV content to contaminate the shot, or would it all be 'absorbed' as contended by Benji.