1) The term fully multi-coated is somewhat misleading and has been abused by too many "marketeers". There are any number of different coatings that can be used on a piece of glass, and in some cases, depending on the type of glass, a single layer coating can actually have a lower reflectivity than a multi-layer.
2) In the case of a cemented 2 element Barlow, an anti-reflection coating has only one effect - it increases the light throughput. It does not have any effect on the contrast, ghosting, light scatter or sharpness. Adding just a single layer to each of the two air-glass surfaces increases the light transmission by 7%. Adding a multi-coating increases the light transmission another 2% to 2.5%. The latter is probably not all that noticeable.
3) The Astro-Physics Barlow is a 2 element cemented multi-coated lens of negative 127mm focal length. It does indeed work as a Barlow and will work with any eyepiece (I can't think of one reason why it would not). It's job is to merely increase the effective focal length of the main objective. Being that it has a long focal length, it diverges the off-axis rays far less than that of a "shorty" barlow.
4) In case you are wondering what kind of coating is on this Barlow, the image shown here compares 4 different coatings: you see the reflectivity of an uncoated, single layer coated and two multi-layer coated surfaces. The AP Barcon as we presently deliver is shown in the last two images. The Barcons have between 0.25% to 0.5% average reflectivity. The reason for the spread is that it is very difficult to control that last 0.25% in actual production runs, although most of the lenses that we have received in the recent runs have been of the very lowest reflectivity.Roland Christen, ASTRO-PHYSICS