The Mishna (Nazir 57) describes a case of a witness who sees one of two nezirim become tamei and is not sure which one it was. Putting aside for the moment what the nezirim faced with this safeik should do, the gemara discusses why there is a safeik in the first place. The rule of thumb when it comes to sfeikos is that safeik tumah in a reshus hayachid is tamei, safeik tumah in a reshus harabin is tahor. Since this case involves 2 nezirim + the observer, it should be considered safeik tumah in reshus harabim and both nezirim should be tehorim. The gemara answers that the case must be where the observer was not immediately proximate to the two nezirim but just saw tumah thrown in their direction.
Tosfos asks: If the observer is not proximate to the nezirim, then isn’t this a case of safeik tumah in a reshus hayachid and both nezirim should be tamei? Tosfos answers that the rule of safeik tumah which is derived from the parsha of sota does not apply where we are forced to draw two mutually exclusive conclusions. The fact is that only one nazir became tamei. The halachic rule of safeik tumah b’reshus hayachid being tamei would force us into the counterfactual conclusion that both nezirim are tamei.
Tosfos’ pilpul is not over. It we throw out the rule of safeik tumah b’reshus hayachid being tamei, then why not also throw out the rule of safeik tumah b’reshus harabim being tahor? What then was the gemara’s question to begin with? Tosfos answers that the rule of safeik tumah b’reshus harabim being tahor is not based on a gezeiras hakasuv from parshas sota but is based on chazaka. Each individual nazir has a chezkas tahara. Even if both chazakos mutually exclude each other, until we can prove which nazir became tamei, the chazakos still stand.
Tosfos repeats in many other places this idea that safeik tumah b’reshus harabim is not derived from a gezeiras hakasuv and it really deserves a separate discussion (most of ther first section of Shev Shamytza deals with this topic). For now, I just want to point out that R’ Chaim had a different approach to this whole question. R’ Chaim reportedly answered that Tosfos is right – based on the gemara’s answer this should be a safeik tumah b’reshus hayachid. So why does each nazir not do a tiglachas tumah? Let’s say the safeik is whether these nezirim came in contact with a dead body – why is each one not assumed to be tamei meis?
There are two ways to understand safeik tumah b’reshus hayachid being tamei. One way is to assume that the rule reveals the facts on the ground: each nazir actually touched the dead body in question. Tosfos took this approach and rejected the rule of safeik in this case because it leads to two mutually exclusive outcomes. R’ Chaim suggested a different approach. In actuality we remain uncertain whether either nazir touched the dead body. However, the halacha has created a categorically new type of tumah that applies to situations of safeik. What is gained by this approach is that the new rule can encompass even circumstances that lead to mutually exclusive outcomes. However, since the tumah we are dealing with is a categorically new entity, it is not related to tumas meis or any of the other tumas which a nazir must do tiglachas for. That is why the Mishna's conclusion is that neither nazir must do a tiglachas.
Whether R’ Chaim’s model or Tosfos’ model of how safeik tumah brh”y works is correct seems to be debated by the ba’alei haTosfos in other sugyos, but enough for now.