Amorgos is an island for walkers with a good, really good, black and white, specially drawn walkers' map. On it Agia Anna is described as suitable for naked bathing.

From the bus stop at Agia Anna there is either a concrete path down to the left to the textile beach with the chapel on it (no nudism here). About 50m down this path one can go right and scramble across the rocks to a lovely ledge where an afternoon naturist set hangs out. It is however overlooked from the car park and is at best 50% nude.
Alternatively you can go right from the car park past the canteen and down the zigzag path to the beach facing the rock shelf. It is heavy gauge rocks but on a still day it is like a scene from the big blue. Probably only 25% nude at best and overlooked again from the car park.

The "real" naturist beach, named Kambi is about 250m further to the right in a scramble along a very rocky path at the water's edge.

A few minutes of scrambling to the left of the ledge mentioned above, where the "afternoon naturist set" go, (in the direction of the chapel, in other words left if you are facing the sea) and you will find a beautiful naturist cove. It is not visible from the car park, the rocks are flat and quite comfortable, it has sun late (for Agia Anna) and there is a great view of the monastery.

2008: Areport suggested that the beach was crowded and no nudes there, but that snorkelling was excellent.

2017 July: A report states that the beach is in pristine condition and has very few visitors. During one visit there was one couple and a family, all nude. During a second visit there were about 18 people there, with 70% nudist.
A report from August states that the beach is a great place to relax and swim nude, few people, very relaxing place with a great view of the monastery (I think this refers to Kambi as you can't see the monastery from Agia Ana beaches)

2018 June: Both bays at Agia Anna are small. 10 people on each would make them crowded. The small bay to the left, next to the chapel was all textile, the next bay accessed from under the cafe was 80% textile and the only nude couple covered up as the beach filled up. On the rocks between the two was a single nudist. There was a steady stream heading on to Kambi along the rocks. Beautiful clear blue water and fantastic snorkelling.

2021 July: very small crowded beach. 100% textile.