This question is prompted by widespread speculation that the Labour party might split if and when Jeremy Corbyn becomes its leader, but I am also interested in general terms.
When parties split, if the breakaway group is small no one disputes that they are a breakaway group and that the other, larger group get to keep the party name, funds, membership list and so on. An example is when the the "Gang of Four" left the Labour party to form the SDP in 1981.
But what if the group "leaving" and the group "staying" are roughly the same size? Both might have plausible claims to be the "true" Labour party. Both might reasonably claim to be the ones who are staying and claim that it's the other guys who are leaving them. You could also get intractable problems if the split among MPs is very different to the split among the party members. Bringing further variables into the situation, such as what happens regarding paid party officials and trade unions who provide funding for the party, makes it even more complicated.
How could such a split be handled fairly? Are there any precedents for this situation, or laws dealing with it?