Yep, I was right, it's a routing issue. The fix is a little tricky, it's like this:
1. Run an elevated cmd.exe (press start, type cmd, press ctrl-shift-enter, approve UAC prompt). Make sure cmd.exe window says "Administrator:" in the title bar.
2. Run "route print" and then in the first section "Interface List" find the leftmost number for the line that corresponds to the "Hamachi Network Interface." For me this was 20.
3. Run "route delete 0.0.0.0 IF 20" (replace 20 with your number from step 2).
That's it. Before I did this, I had a line in the "Persistent Routes" section that listed the hamachi interface handling 0.0.0.0 with default metric, even though the line in the "Active Routes" section listed the metric as 9256. I'm not sure what all that means, but Hamachi still works fine after doing this (I'm not sure why it would need a catch-all 0.0.0.0 route anyway).