Every once in a while I’ll play someone (IRL) who will just mirror my moves (for example if I play at R16 they might play at 4C), and so forth.
This usually ends with me saying something like “oh c’mon lets just play normally” or something.
So how would I actually continue to play against someone mirroring my moves without rejecting the game in such a way where I win the game?
clever way, play in such a way that the person making the first move gets to capture and that means the second player can’t (because the groups are touching)
lazy way, just play tengen
Also take a look at SL’s page on countering mirror go.
Edit: @aheavyburden: We’re linking to two different pages above.
In first 
One trick is to make a ladder, something funny happens if you can get the ladders to meet in the middle 
Well, one of the common ways to play mirror is to be black and start at the tengen. That way, your opponent cannot break the mirror by playing the only unmirrorable move, and funny enough it is not so easy to set up a usefull mirrored ladder that does not go through tengen in the beginning of the game.
So I do not think it is as easy as it may sound. (or I am just bad at Go, always a real possibility :D)
The thing is nobody wants to mirror forever (if playing seriously) and are just waiting for some “slack” move that sets up some unmirrorable ladder to stop mirroring anyway and take advantage elsewhere. So I guess one of the ways (if you are white) is to just wait it out (since you have komi) stare deeply in your opponent’s eyes and completely destroy their confidence.
AdamSavage
It’s probably not advantageous to play mirror go as black for too long. Once the opponent notices mirror go you’re kind of giving them control, plus you’re behind by komi as well so you can’t really play mirror go long term unless tengen becomes useful at some point. You could set up two avalanche Josekis or some new alphago 3-3 josekis that get complicated and hope your opponent makes a mistake if they give up on playing mirror I’ve a feeling Michael redmond mentioned something about mirror go when reviewing one of the alphago games, just because a few of the moves at the start looked like mirror go, but it was 5 moves in or something.
Well, it’s quite an interesting topic really, and I guess up to ones preference, but I disagree. If I were to play mirror, I would chose black. With white it is easy to counter, tengen being the move nobody can mirror. Or setup a ladder first and then play tengen… Lots of options.
If you take that move first it is up to you when to break the mirroring (and most ladders now favour you). And as said, in “serious” games I believe nobody wants to mirror forever. You wait for the right moment to come up with your own attack.
5 moves is not mirror go there was one game when “master” was being tested online when some pro tried to mirror him/she/it. If I am bored tomorrow I will find the kifu somewhere…