Simultaneity in Minkowski spacetime: from uniqueness to arbitrariness

In 1977, Malament proved a certain uniqueness theorem about standard synchrony, also known as Poincar\'e-Einstein simultaneity, which has generated many commentaries over the years, some of them contradictory. We think that the situation called for some cleaning up. After reviewing and discussing some of the literature involved, we prove two results which, hopefully, will help clarifying this debate by filling the gap between the uniquess of Malament's theorem, which allows the observer to use very few tools, and the complete arbitrariness of a time coordinate in full-fledged Relativity theory. In the spirit of Malament's theorem, and in opposition to most of its commentators, we emphasize explicit definability of simultaneity relations, and give only constructive proofs. We also explore what happens when we reduce to "purely local" data with respect to an observer.