I am attaching a picture of the beam on a IR card as well as the tapered end.
This fiber, 50 um core 125 um cladding diameter, was tapered to 12-13 um cladding diameter, so core diameter became 5 um. The input beam is 1064 nm, randomly polarized, Gaussian with 1/e2 diameter of 5 mm, focused onto the fiber's untapered end by NA 0.22 10x objective.
I observed a similar doughnut shape with a 532 nm linearly polarized beam.
Maybe this is trivial and someone else who observed this already knows the answer.