Magento forgot to add the formkey block to the following template:
app/design/frontend/base/default/template/checkout/multishipping/addresses.phtml
Magento should fix this in an update of the SUPEE-9767 patch (and the CE 1.9.3.3 release), but in the meantime, the issue can be circumvented by adding the following line just somewhere between the and tags<form> and </form> tags of this template and all overrides (if any) of this template:
<?php echo $this->getBlockHtml("formkey") ?>