The most common case of gender mismatch is when grammatical gender is masculine and actual gender is feminine (человек, преподаватель, рабочий, сотрудник and so on).
In such case, adjectives agree in grammatical gender and verbs usually agree in actual gender:
Хороший преподаватель сказала...

What about сирота, it is marked as общий род in a dictionary, which means you can agree other words in both masculine and feminine:
male: хороший сирота шёл
female: хорошая сирота шла

Referring to a male person with grammatically feminine nouns is rare enough. I can think of such words as личность, персона. (...задница, сволочь, скотина... ooh, or not so rare )
It seems they always require agreement to grammatical gender.
"Он пришел?"
"Нет, эта ленивая личность сегодня не явилась."