If all plants try to stay in-phase, how do power plants interact with other plants further away?
All the generators in the same power grid are in sync (which is not the same as being in phase). Only nearby generators are in phase. For example, when a new generator is connected to the grid, first the new generator is brought in sync with the rest of the grid, and connected when there is no phase difference between them.
Apart from that, power plants do not just connect/disconnect generators at their will, there is a National Dispatcher bureau in charge with operating the national grid (at least here, in Ro). The dispatcher tells when to connect or disconnect generators, when to re-route the power across major HV lines, etc.
The power plants do not interact much with each, they obey the national dispatcher. There is some communications along the nearby power lines, but that is mostly for automated protections. Power plants do not negotiate with each other regarding the levels of power they deliver (which affects the phase).
The discipline when maneuvering the national power grid reminds of military discipline, or of the discipline of a control tower from aviation. This is because a faulty maneuver may damage equipment that cost millions, or even worst, may leave a huge area (if not an entire country) without electricity, and for such an outcome the damage/consequences are incalculable.
All the maneuvers are described to the finest detail, all steps are followed to the book, all possible outcomes (and how to react to that) is known by heart. During a maneuver, each step is executed with confirmation coming from both the human operators and the machines/instruments. Often there are people in the field to look at the equipment, and to visually confirm each step was indeed executed successfully.
The discipline and professionalism of the people involved with the national power grid is hard to imagine without witnessing them at work.