Phlegmatically means in a calm way, without getting excited or worried. It tells you how someone speaks, waits, or acts — slowly, evenly, not jumping. You might hear it in writing more than in speech; it is a long, slightly old-fashioned adverb, and many people would just say calmly or without fuss. The feeling can be admirable (they kept their head) or a little cool (they barely reacted). Same personality picture as phlegmatic: the old humour story is in the background, but the living use is about not being easily stirred.