The first time that I came across the term besom pockets was when the tailor, who was making my formal dinner jacket, (tux) asked me the same. He explained that besom pockets, are pockets that are cut into a garment, rather than being sewn on. They generally have reinforced edging or piping along the slit of pocket as seen here on my jacket.
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I have since heard that besom pockets have no flap. My assumption on a non besom pocket is that it is a non integral pocket, similar perhaps to the rear pockets on a pair of jeans. However, when it comes to the minutiae of the details, George is an authority. My intention is to have the suit made, as is. Great thread by the way, you would get on well with my tailor.
Thank you. I expect I would. I enjoyed chatting with my tailor, while thumbing through bunches of hundreds of swatches of beautiful cloth.