We don't know, as it has never happened.
To speculate, the obvious thing would be to make some kind of agreement where the third party Vice President caucuses with one or the other party. For example, VP Bernie Sanders might continue to caucus with the Democrats in that circumstance, as he does as Senator. Same thing for Angus King. If Bill Weld had been elected as VP to Gary Johnson, Weld would have been likely to have caucused with the Republicans.
Another possibility would be that a group of centrists from the two parties might band together. A Gang of 51 if you will. At that point, they could effectively write the rules. Note that the nuclear option precedent clearly established that a simple majority can rewrite the Senate rules that aren't constitutionally mandated.
Two groups of extremists could work together to change the rules. It's unclear what the basis for their cooperation would be, but there's nothing legally preventing it. All they need is a majority of the votes.
Would they have to elect a President Pro Tempore that would actually do work, and have that person act like the Majority Leader does now?
More likely than that would be for the VP to run the Senate. The rules don't support that now, but it seems how the system was designed in the constitution. The VP is supposed to be the leader of the Senate. When the VP is not available, the President Pro Tempore would be in charge. If the VP just asserted those rights, the courts might uphold them. They don't like to interfere in the Senate's everyday operation, but in a crisis, they might.