Opps should have thought this through!
I think I would probably use icing and a round tip to fill in the grouted areas. I wouldn't think you'd really have to wait any time to do it after you placed all the tiles on the cake.
What are you making the tiles out of? What are you icing the cake with?
I would think if your icicg the cake in bc and the tile out of fondant, if you roll the fondant real thin you probably won't need to do grout.
If your making the tiles out of isomalt, they would be certain thickness and you could probably grout with bc in a piping bag.
Don't quote me, just a thought.
I tried this once on a small cake. I cut out pieces of fondant and "glued them on with icing leaving a space of about 1/8" between the pieces. Then, instead of covering all the pieces with icing and then wiping it off leaving the icing in the grooves (like you would with real tile and grout) I used a round #4 tip (I think) and piped into all of the grooves, then I went around with a paintbrush to smooth and flatten the grout lines and removing any excess. I didn't wait between doing the tiles and the grout.
I agree with crabbabs re using the paintbrush, the effect is a lot tidier
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