"Who would be prime minister?" seems to be a bone-of-contention at present, were an alternative government to be formed within the present house.
As I understand it, the phrase "prime minister" was originally somewhat of an insult, suggesting that one government minister had ideas above their station of collective cabinet responsibility. Though this may have later changed into a codified office?
However, there is also the office of "First Lord of The Treasury" which seems to have been long synonymous with the Prime Minister, and may actually be the office to which many apparently prime ministerial functions attach. There are also institutions such as "Prime Minister's Question Time" which might be tricky were there no nominated Prime Minister.
Ignoring for a moment if any of the current characters would be willing to share such an office or to leave it vacant, are there major barriers to the functioning of parliament or government, with these roles unfilled.
To be clear, I'm asking for clarity on which titles traditionally afforded to the person we call Prime Minister are offices, carry responsibilities, and functions, etc, and the extent to which were such leadership offices vacant (or shared on various bases) a government could function from a constitutional standpoint.