If two clocks are stationary with respect to each other, then there is no ambiguity about where the midpoint between them is. Then you just do something like, when your clock hits noon, fire a projectile at a fixed speed toward the other clock. If the projectiles meet at the halfway point, then the clocks are synchronized.
And there is no relativistic funny business involved, because the clocks are stationary with respect to each other. There's no difference of viewpoint as to whether the projectiles met at the halfway point, or where the halfway point was, or even how far off from the halfway point they met (and therefore how far off the clocks are from each other).
This is the argument used in my relativity class to show that you can synchronize clocks that are stationary with respect to each other. (You have to be able to do that to construct an inertial frame of reference, that is, to be able to determine what time coordinate some event occurs at, no matter what spatial location it occurred at.)
And there is no relativistic funny business involved, because the clocks are stationary with respect to each other. There's no difference of viewpoint as to whether the projectiles met at the halfway point, or where the halfway point was, or even how far off from the halfway point they met (and therefore how far off the clocks are from each other).
This is the argument used in my relativity class to show that you can synchronize clocks that are stationary with respect to each other. (You have to be able to do that to construct an inertial frame of reference, that is, to be able to determine what time coordinate some event occurs at, no matter what spatial location it occurred at.)