According to this answer tunneling probability depends, among other things I don't know, on the length of the barrier.
Due to length contraction when going at relativistic speeds, it seems it should be theoretically possible to contract lengths arbitrarily by going at an arbitrarily high fraction of the speed of light. I suppose that means the probability of tunneling can get arbitrarily high if we just go fast enough.
At what point does the probability of a particle (say, a proton) tunneling through, say a planet or a star, significant. Let's say >10%?
Is this a case where quantum mechanics and special relativity still play nicely together, or not?