In actual fact:
— To be nominally but not actually independent.
— No one actually saw the shark.
— Large meteorites actually come from the asteroid belt.

synonym: really.

Roget 494: truly etc. adj.; verily, indeed, really, in reality; with truth etc. (veracity) 543; certainly etc. (certain) 474; actually etc. (existence) 1; in effect etc. (intrinsically) 5.    exactly etc. adj.; ad amussim [Lat.]; verbatim, verbatim et literatim [Lat.]; word for word, literally, literatim [Lat.], totidem vervis [Lat.], sic, to the letter, chapter and verse, ipsissimis verbis [Lat.]; ad unguem [Lat.]; to an inch; to a nicety, to a hair, to a tittle, to a turn, to a T; au pied de la lettre [Fr.]; neither more nor less; in every respect, in all respects; sous tous les rapports [Fr.]; at any rate, at all events; strictly speaking.    ... show more

Used to imply that one would expect the fact to be the opposite of that stated; surprisingly:
— You may actually be doing the right thing by walking out.
— She actually spoke Latin.

synonym: in reality.

Roget 1: actually etc. adj.; in fact, in point of fact, in reality; indeed; de facto, ipso facto.   

At the present moment.

As a sentence modifier to add slight emphasis.