Thanks for your answer! Indeed I forgot to mention that I am using
bwa mem version 0.7.17. Thank you for pointing out the -a option. This indeed leads to all reads mapping twice (once in each part), in the manner I would expect. One of the alignments gets a 256 flag. But when I filter the reads with the 256 flag out, I get the same result as before. The distribution might be uneven for stochastic reasons but I cant test that because the outcome is absolutely reproducible, not randomly generated in reach run.
I cannot share the data, because it's not mine but I can share a screenshot of the alignments from IGV. On the top you see the outcome with the -a option. At the bottom without it.