My response was to G Jenista's suggestion of a float bridge as a solution. The Confederate Bridge is a concrete box beam bridge supported on piers above the water.

The Bering Strait is a very trecherous place. Huge seas, moving ice flows and strong currents are among some of the challenges. I have seen a documentary on a proposal for an over the water bridge that addresses these issues. Even if the bridge is built, the weather would make a crossing impossible much of the time. I suspect the weather is what may make a tunnel a more practical solution in this location.

I appears that a crossing could be done but what would it be economicaly or even politically feasable...I don't know.