This paper investigates what sort of security can be retained by the most efficient (namely, rate-one) AE schemes like OCB under the release of unverified plaintext (RUP). At CT-RSA 2016, Chakraborti et al. have presented an impossibility result, which says that any rate-one AE scheme cannot ensure INT-RUP, a strong integrity requirement under RUP. In this paper we show that any rate-one AE scheme cannot satisfy PA2 (plaintext awareness 2) either, a strong privacy requirement under RUP introduced by Andreeva et al. at Asiacrypt 2014. Given these impossibility results, we relax the security requirements and identify new notions of tag-PA and tag-INT. The new notions are strictly weaker than PA2 and INT-RUP yet have considerable significance in the practical sense. In particular, tag-PA is strictly stronger than PA1 defined by Andreeva et al. at Asiacrypt 2014. Unfortunately, OCB is neither tag-PA nor tag-INT. We present a new rate-one AE scheme which is both tag-PA and tag-INT. The new scheme is essentially as efficient as OCB, consuming just one extra call to a block cipher.