The wrapping apparatus and the wrapping method according to the invention wrap an object (B) in a chamber (Ch) into a web (20) of wrapping material. The web (20) is moved from a reservoir outside of the chamber (Ch). A tag applicator (5) applies a tag (50) on the web (20) in an application position (AP) which is between a wrapping material reservoir (30) and a wrapping material inlet (WI). A tag reader (4.2) monitors whether or not a tag (50) is conveyed in the chamber (Ch) along the tag reader (4.2). If no tag (50) is detected, an alert is generated that the object (B) is not properly wrapped.
