This answer elaborates on the answer given by stackexchange contributor Dale
The case of a closed Minkowski space-time is discussed by Olaf Wucknitz, in a 2004 article titled: Sagnac effect, twin paradox and space-time topology - Time and length in rotating systems and closed Minkowski space-times
Wucknitz argues that understanding of the Sagnac effect carries over to understanding of a closed Minkowski space-time.
The Sagnac effect is mostly known for its essential role in ring interferometry.
To explain the Sagnac effect I will use the case of a series of relay stations, positioned around the equator. Let's say the equator of the Earth. The number of relay stations is arbitrary, let's put it at twelve. The relay stations are in radio contact. These relay stations proceed as follows: they start two counterpropagating relay transmissions, consisting of pulses. Let's say twelve pulses. Each time a pulse is received it is retransmitted to the next relay station.
Each pulse train, the clockwise and the counter-clockwise, has the same amount of pulses. The timing of the pulses is adjusted as follows: the measured time interval between the twelve pulses clockwise propagating pulses is the same, and the measured time interval between the twelve counterclockwise propagating pulses is the same.
Under those circumstances the clockwise and counter-clockwise pulse trains will not have the same time interval, as measured by the relay stations. Here is why: as a matter of principle the two counterpropagating pulses are moving at the same speed: the speed of radio transmission. The relay stations are co-rotating with the Earth, clockwise as seen from the south pole. Each pulse takes some time to travel from one relay station to the next, so in the time between a relay pulse being emitted and a relay pulse being received the relay stations have moved in the clockwise direction. This lengthens the measured time interval between the pulsed of the clockwise pulse train, and it shortens the measured time interval between the pulses of the counter-clockwise pulse train.
Note especially that this lengthening/shortening is enforced by the constraint that the clockwise and counterclockwise pulse train must consist of the same number of pulses. You have to count the number of pulses.
Returning to the relay stations positioned around the equator: the difference in measured time interval between the clockwise and counterclockwise propagating pulses indicates the rotation rate of the Earth. More specifically, it allows you to identify the non-rotating cordinate system.
Again: the principle that gives rise to the Sagnac effect is that in all directions of propagation the speed of electromagnetic radiation is the same.
The Sagnac effect comes into play when you close a loop. In Minkowski space-time, when you close a loop very interesting things happen.
If you don't close the loop then you don't get the Sagnac effect. Imagine a setup where the loop isn't quite closed, but that instead there are two stations that do not relay to the next station, but they bounce the pulse back. Then you get the standard Einstein synchronisation. When you do close the loop, and you compare clockwise and counter-clockwise propagation, you get access to information that you otherwise would not have access to.
A closed Minkowski spacetime
Olaf Wucknitz argues that in a closed Minkowski space-time one can set up the same procedure as in the above explanation of the Sagnac effect. That will establish a universal reference frame.
In general: in Minkowski spacetime closing a loop is significant. By contrast: in Newtonian space & time closing a loop isn't particularly interesting. But in Minkowski spacetime closing a loop makes all the difference.