But what would LAN tell you if you called them? "No, we're inept at this" is probably not the official company line.

When AY changed to a more restrictive policy (previously, they tagged the bags to any interlining airline even on separate PNRs), they stated that the reason for the change was that any mishaps that happened with the luggage would then be AY's responsibility to reimburse. So it's cheaper for the airline not to tag bags through, and some airlines may want to avoid doing that if they're not that worried about what that does to their customer experience. When it's all on the same airline, that's not a consideration, so suddenly the computer has no difficulty in the incredible feat of printing out any string of three-letter codes in a row.