Final answer:
Greenland is the country that, while controlled by Denmark, voted for self-rule in 2008 but Denmark still handles its foreign affairs, security, and financial policy.
Step-by-step explanation:
The country that is controlled by Denmark and voted for self-rule in 2008, yet still has its foreign affairs, security, and financial policy managed by Denmark, is Greenland. Greenlanders voted in favor of the Self-Government Act, which was implemented on June 21, 2009.
While the Greenlandic government takes care of most domestic affairs, Denmark retains control over foreign affairs and defense and provides a sizable subsidy to Greenland's economy. This relationship allows for a unique form of autonomy within the Kingdom of Denmark.