TV Tropes calls the plot device “The Man Behind the Man”.
Shakespeare does not use this device. The device works by withholding information from the audience in order to surprise them by revealing it later, but Shakespeare prefers to use dramatic irony by revealing information to the audience that is withheld from the characters. I’ll give a couple of examples to show how this works.
First example. In Macbeth, the witches have prophecied that
Macbeth shall never vanquish’d be, until
Great Birnam wood to high Dunsinane hill
Shall come against him. [IV.1]
A dramatist using surprise would show us Birnam wood moving to Dunsinane hill, and then reveal that it is actually Malcolm’s army carrying branches. But Shakespeare does it in the opposite order, putting the revelation first:
Siward. What wood is this before us?
Menteith. The wood of Birnam.
Malcolm. Let every soldier hew him down a bough,
And bear’t before him. Thereby shall we shadow
The numbers of our host, and make discovery
Err in report of us. [V.4]
This means that the audience have all the information needed to appreciate the irony of the situation and anticipate Macbeth’s response to it.
Second example. In Titus Andronicus, Aaron says that the emperor will reprieve Titus’ sons from being executed, if Titus will sever his own hand:
Aaron. Titus Andronicus, my lord the Emperor
Sends thee this word, that, if thou love thy sons,
Let Marcus, Lucius, or thyself, old Titus,
Or any one of you, chop off your hand
And send it to the King: he for the same
Will send thee hither both thy sons alive,
And that shall be the ransom for their fault. [III.1]
A dramatist using surprise would show us Titus losing his hand and then reveal that Aaron was lying and Titus’ sons are already dead. But again, Shakespeare does it in the other order, so we can experience the horror and pity of watching Titus mutilate himself for nothing:
Aaron. [Aside] If that be call’d deceit, I will be honest,
And never whilst I live deceive men so;
But I'll deceive you in another sort,
And that you'll say ere half an hour pass. [III.1]
Almost the only case in Shakespeare where he withholds a major piece of information from the audience is in The Winter’s Tale, and this is so unusual for him that critics have speculated that the play originally had a different ending and the last act was rewritten, as discussed in this answer.
It is a remarkable instance, the only one in Shakespeare or perhaps in the whole drama of the period, of the playwright’s concealing so material a circumstance from the audience.
Frank Kermode (1972). ‘Introduction to The Winter’s Tale’. In Sylvan Barnet, ed. The Complete Signet Classic Shakespeare, p. 1497. New York: Harcourt Brace Jovanovich.