I feel like socialist wing of the Democrats has both the insistence and the hubris required to split off, especially after another particularly dire primary squabble with the party establishment. Whether they would actually be a serious electoral proposition is another question entirely. Under FPTP, it would likely require some kind of negotiation with the Democrats in agreeing where to run and not run candidates. However, if this division was as acrimonious as it would have to be to cause an outright exodus, it might not be easy to get the two parties to talk.
I could see them sharing the same primaries and being able to have amicable turf wars and being able to negotiate coalitions when the GOP wins less than 50% of the power but a strong plurality. My guess is that this arrangement would allow centrist or insufficiently ideological candidates to run as stereotypical conservadems on a platform of nuance on health care, ending the payroll tax max, taxation on pensions, while also running against taking up more civil rights causes and going to the center on women’s rights, self defense rights , and gay rights. The entire space between Charlie Crist and Dan Lipinski.
Then the rest of Democrats could either join that party or feel more comfortable running on government guarantees of training, health care, retirement, and balanced trade and employment. Also on drug and gun reform, civil rights, reproductive rights and taking a much more populist stance on pollution.