...you don’t actually need a tether here. These planets are co-orbiting each other, and will naturally stay together indefinitely (barring interference from another planet or star.)
The main effect of this is tidal locking; each planet will always be facing the same side towards the other. This means tides will be affected only by the sun, as there is no moon moving around the planet, and will probably be much weaker. (Someone correct me if I’m wrong.) however, because the planets are tidally locked to each other, and not to the sun, they will both still have a day/night cycle, which will be the same on both planets.
If you’re wondering why tidal locking doesn’t happen on earth, it does; the moon is tidally locked to earth. This is why we never see the dark side of the moon. However, because earth is so much bigger than the moon, earth is not tidally locked to the moon.
Similarly, the earth is not tidally locked to the sun because the two are very far away. These planets are much closer together, so the near sides will be pulled observably harder than the far sides, causing tidal locking.
Also, Since these planets are larger that earth’s moon, eclipses will probably be bigger and more frequent.