Presumably due to the strange collision boxes of anvils, it is possible to walk through them.
To reproduce, place two adjacent anvils facing different directions (perpendicular) and push against the border of the two. With a bit of turning, you can easily walk into the anvils.
There's also a second method I found that is more difficult to reproduce/describe, so yay pictures. (requires more turning)
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I have also found that if you just walk into an anvil, the screen gets a bit jerky.
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Confirmed. Also, once between the anvils, you can walk into one of them.
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Problem with anvil hitbox, more details in MC-1669.