Ask where exactly the border between Germany and Luxembourg runs, and for a long stretch the correct answer is: nowhere. Along the section formed by the Moselle, the Sauer and the Our there is no line. There is a shared surface.
This is not cartographic sloppiness but law in force. The three rivers are joint sovereign territory of both states. German sovereignty does not stop in midstream but at the far bank, and Luxembourgish sovereignty likewise stops at the German bank. Both states are competent on the same stretch of water, across its full width.
Lawyers call this a condominium: a territory over which two or more states exercise sovereignty jointly, without dividing it. It is rare, because it forces the administrations involved into constant coordination.
What the two states share here is not the area but the competence over it.
— TerraTally · Borders desk
Where it comes from
The origin lies in the Final Act of the Congress of Vienna of 1815. In substance it says that rivers, where they form the border, belong jointly to the adjoining powers. The border treaties of 1816 between the Kingdom of the Netherlands, which Luxembourg belonged to at the time, and Prussia put that sentence into effect for this section.
What is remarkable is what did not happen afterwards: the arrangement was never undone. Two world wars, Luxembourg becoming an independent state, the founding of the Federal Republic, European integration, any of these could have been an occasion to finally divide the rivers. The joint sovereignty stayed. A border treaty of 1984 settled points of detail without touching the principle. The arrangement is older than both states that apply it today.
As long as nothing happens on the water, nobody notices. It gets interesting with anything that requires jurisdiction: who permits a jetty? Which police force is responsible when something happens on the water? Whose law applies on board a vessel in midstream? On a divided border each of these questions has a simple answer, namely the side you are on. In a condominium that answer is missing, because there are no sides.
For such cases there are agreements and joint bodies in which authorities from both countries settle who does what. The principle stays inconvenient all the same: two administrations have to agree about the same surface instead of each deciding on its own side. Anyone building on these rivers, or seeking a permit for something there, ends up dealing with both. The effort has not disappeared, it has moved: off the map and into the coordination.
That is precisely why condominiums are rare. Most border rivers in the world are divided, either down the middle or along the deepest channel, the thalweg. Both produce a line, and lines are administratively convenient: they can be drawn on maps, translated into responsibilities and measured again in a dispute.
The Moselle, the Sauer and the Our are the exception. For more than two hundred years nobody there has bothered to draw a line. And it works.