You're only doing half the work now :-)
You can make it work like this though:
++$count{$1},pos($genome)-- while $genome =~ /\G(..)/g
It's interesting that this is slightly faster than dropping the \G.
This however turns out to be a lot slower (seems to do a complete linear search for the proper string start every time):
$genome =~ /./g;
$count{$1}++ while $genome =~ /(.\G.)/g
update
This however means that the lookahead version can be speeded up by about 5% too:
$count{$1}++ while $genome =~ /\G(?=(..))./g
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