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This was actually originally mooted in the 19th century and occassionally comes up as a possible investment project. The current suggestion is to build to a bridge to the Isle of Man from Ireland and then onto the UK mainland. I don't know if it would ever by economically feasible or viable, but there are longer bridges between Danish Islands and of course there is the magnificent bridge crossing between Denmark and Sweden.
You would then have no non land border within the EU. BTW I would love to see it!