Forgive me if I have this wrong, but....
What happens in bwa mem is that often one gets the primary alignment, and lots of much shorter secondary alignments, many of which are hard/soft clipped. These secondary alignments are essentially local/split alignments wrt to the read, and are often not what SNP discovery projects are interested in.
GATK ignores everything except the primary alignment.
However, where reads do not have a good, long primary alignment, I am guessing it's possible that the short/clipped/local alignment is marked as the primary alignment and is taken into consideration by GATK when SNP calling.
For example, if one has PhiX in the reads, but not in the reference database, then one can get short/clipped/local alignments of the PhiX reads against the reference genome. These are clearly false alignments. However, they may be marked as primary alignments, and GATK may use them to call SNPs.
I think that was the focus of the question....