What are "anx9c1s6 type fares", as seen on the following screenshot I got from britishairways.com while making a booking?
(Googling yields zero hits.)
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I spoke with BA over twitter.
The code would relate to the rules and restrictions on the ticket. If booking via our website, these would be advised at the time of booking. Nearly all flights would have underlying fare types such as this, and as you point out, 'A' fares would usually be restricted First class fares. These would normally have fees attached to change the booking, and may have a fee for cancellation or be non-refundable.
So as suspected, the A is restricted first class, but the rest is related to your specific booking, and they couldn't give more information without more information from your side.
As fare codes, after the first letter, are airline specific and BA does not share the full list with the general public, you may never find the full explanation. You worked out much already by working out you need to stay a Saturday night.
It is a long and complicated code but those are not unlikely, even though they are rare.