Question:I, a kohen, turn around at the end of Birkat Kohanim when the chazan starts Sim Shalom. Recently, a chazan chanted a tune between Birkat Kohanim and Sim Shalom. Was that proper? Were we supposed to turn around when he started chanting or when he said Sim Shalom?

Answer:The gemara (Sota 39b) indeed says that kohanim should not turn around until the chazan begins Sim Shalom. Therefore, it seems that you should have waited until he actually started Sim Shalom, as an introductory tune does not have halachic standing. However, the matter deserves a better look.
Rashi (ad loc.) describes the end of Birkat Kohanim as follows: the congregation finishes saying Amen to the last beracha, the kohanim turn around and close their hands, the chazan starts Sim Shalom, and the kohanim start reciting "Ribono shel olam." His order places turning around after Amen but before Sim Shalom (i.e., in your case, you did not have to wait). How could Rashi contradict an explicit gemara? The Maharshal (ad loc.), based on Rashi, says that the gemara means that the time for Sim Shalom must have come, i.e., the congregation must have completed answering Amen.
While Tosafot (Sota 39a) and the Ran (Megilla, 16a of the Rif’s pages) quote Rashi without comment, the Rambam (Tefilla 14:6) and the Shulchan Aruch (Orach Chayim 128:15) bring the halacha with the simple reading of the gemara – when the chazan starts Sim Shalom. There is no indication from their wording or the sources that the Beit Yosef and commentaries cite that their intention is the Rashi/Maharshal approach.
Let us see if there is halachic logic to have to wait literally for the beginning of Sim Shalom. The apparent logic for the kohanim not to turn around immediately is that they should not rush to finish their job before Birkat Kohanim is totally finished, perhaps thereby showing disrespect to the blessings and the blessed (see Birkot Horai 12:(1)). Perhaps, then, Sim Shalom is not necessary, as long as Birkat Kohanim is over.
Does one need to start a new beracha to finish the previous section? This point seems to be at the heart of another halachic discussion. If one did not mention rain in the winter in the second beracha of Shemoneh Esrei until after the beracha, he needs to return to the beginning of Shemoneh Esrei (Shulchan Aruch, OC 114:4). However, the beracha is not considered over in this regard until he begins the next beracha; before this, he can insert the mention of rain where he is up to (ibid. 6). This provides a precedent for the end of one section (e.g., Birkat Kohanim) depending on the beginning of the next (e.g., Sim Shalom).
One might deflect this proof because: 1) Not everyone agrees with that Shulchan Aruch (see Biur Halacha ad loc.). 2) The Mishna Berura (114:31) says that one should start mentioning rain within k’dei dibbur (1-1/2 – 2 seconds) of the beracha’s end. To this, we respond: 1) Not only is the Shulchan Aruch ultimately accepted, but even some dissenters do not say that it is like one started the next beracha but that at that point the mistake is viewed as a nonretractable. 2. The Mishna Berura (based on Derech Hachayim 33:34) only says it is preferable to mention rain right away.
There are also strong indications (based on Megilla 18a) that Sim Shalom is the natural continuation of Birkat Kohanim and may serve as confirmation of the blessing (see Rav Nota Greenblatt in Afikei Torah, pg. 131) and is the appropriate time for the kohanim to commence the second stage of their blessing (see Shulchan Aruch, OC 128:15). Therefore, it makes sense that the gemara means that only after Sim Shalom actually begins should kohanim turn around and say Ribbono Shel Olam.
Kohanim should follow the consensus of poskim (see Magen Avraham 128:28; Mishna Berura 128:70) to not turn around until Sim Shalom starts. Chazanim should not procrastinate or chant before Sim Shalom, which confuses the kohanim and the congregation (see Shulchan Aruch, ibid. 23).