"Theoretically, you could measure the rotation of the earth and call that a day"
This is not theoretical; this is known as sidereal time and is measured to the nanosecond...
Except that "…due to nutation, an actual sidereal day is not quite so constant."
Yeah. Nutation.
Nutation "…is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight…"
Theoretically, you could measure the rotation of the earth and call that a day, but I have no idea how you would measure that down to the millisecond.
As to "Real Time", I don't know about the specific legalities, but the standard goes as follows: everyone uses UTC http://en.wikipedia.org/wiki/Coordinated_Universal_Time which is based off the International Atomic Time http://en.wikipedia.org/wiki/International_Atomic_Time
I too would be interested in how they picked the beginning epoch, i.e. "this moment right here is midnight from which all other seconds shall be referenced against" but in the grand scheme of things it doesn't matter - the only thing that matters is the standard we agreed to measure against.