Question on Awards with Tagging up

[TussAgee11]

I couldn't find this covered anywhere, so I can only assume, but I want to KNOW!

Man on 1st, less than 2 outs. Fly ball to the outfielder, caught. He throws behind R1 to try to double him off. The throw goes into dead ball territory. 2 base award.

Now, do you grant R1 a retouch at 1st as his first base and then award second? Or do you award straight to third and leave it up to him to retouch, for free. Casebook/rules references (couldn't find anything in rules) would be perferred if you have them (I don't own anything but this year's casebook, as I am a new umpire).

Thanks.