"Beviamo il vino nonostante il gusto."
Good point. I guess I would use "despite its taste" if I were referring to a specific bottle or type of wine and "despite the taste" if I were referring to wine in general. I'm not sure if it holds for all native English speakers or if it is a personal preference. Notwithstanding, this is a relatively unimportant distinction that the Italian language might not possess.