For me, making this clear puts in place a strong motivator for folks to be honest with the guard when caught.
"Give me your name, not some false name. If you're unwilling to do so, then we'll kill you and haul your corpse to the administrators, so that if you DO come back, you'll still be exiled AND everyone will still know your name. The only choice you have, is whether you die here and now, or not."