Unless CX has at least one long haul segment on them, they will not reissue it for you.

AA's reissue problem is not just the 16 segment issue. It's the taxes. There are just simply too many to squeeze in the tiny fare calculation boxes. So even if you have something like 10 segments, but if your taxes exceed the space available to put it, they have to hand write it.

It doesn't matter if the tax is attached on a seperate document. Most airlines will not bother except CX and QF where they will read through with a fine tooth comb and a magnifying glass in hand!