What a waste of time this is for RTE News - we all know there will not be a Taoiseach nominated properly and certainly no government has been formed.
This Irish system seems truly weird to me, that the lower house has to nominate the Taoiseach of the day, whereas here in the UK< the leader of the largest party, usually with a majority is appointed Prime Minister by the Queen as they can control a majority in the Commons - why doesn't Ireland do this?
The President appoints the Taoiseach who can command a majority in the Dail, either by majority government, coalition or a confidence/supply agreement?
Not directly pres related, but indirectly I suppose since the procedure of the political developments is highly covered by the media in both countries:
The Irish system isn't totally different to the UK's. In 2010, Labour wasn't the largest party, it still had the possibility of forming a government and was involved in talks until the Con/Lib Dem coalition was confirmed. It was only after several days of coalition talks that Brown went to the Queen to resign and Cameron went to the Queen.
More on Wikipedia: https://en.wikipedia.org/wiki/2010_United_Kingdom_government_formation
The principle is essentially the same between two systems: in the UK the 'would be' PM informs the queen that they believe they have the confidence of the lower house, and so is appointed, and then must maintain that confidence of the house to remain in power (by not losing votes), whereas in Ireland, the 'would be' Taoiseach has confidence by direct vote of the lower house and then travels to the President.
Of course, the PM just exists by convention, whereas the Taoiseach has a role enshrined in the Irish constitution.
Either way, under both systems all the negotiations, votes and trips to heads of state are televised and reported on, just in a different order.