Two tiles were badly broken. One, I replaced easily -- just slid a new one in. Didn't even have to do any additional breaking to get the old one out. But the other one is in the first course, and it's harder:
In the pic, you see the mortar at the front. The tile to the left (barely visible in shadow) is held down by this mortar, and it won't lift up without serious force, which I'm pretty sure will break the tile long before the bond. Toward the back, the thing that looks like a light-colored stick is the foam. The tile's "feet" actually sit on the underlayment (roll roofing?) above the foam. I think that every tile has one of these foam worms holding it up -- both sticking it down and preventing it from sliding.
So I can't slide the new tile in from below because the edge of the tile has to be under the tile to the left, and I can't raise that tile. I could possibly break off a bit of the new tile at the upper left corner to get it started, but then getting the lower edge onto the mortar is just as much of a problem.
Things we've considered:
1) Hire a professional. We have a roofer coming out tomorrow anyway for an unrelated leak, and we'll ask him for a price on this too. He claims to specialize in repairs. Of course it would be good to get more information on just how difficult this is before he comes.
2) Use a circular saw with a masonry blade to cut two slots in the mortar so that I can take out the part under the new tile. Glue it to the bottom of the tile and slide it in. Add enough caulking to make sure it doesn't slide out, though friction seems to be quite adequate to hold in single tiles. (This is in south Florida, so no worry about earthquakes.) If this worked, it would probably look and function very well. But it looks like the blade would have to cut about 4" down from its platform -- basically a blade large enough to cut through a 4x board from one side. Don't have such a saw, though of course small jobs are great excuses for new toys. Even if it worked, it would be a messy job standing on a ladder running a saw to cut though this mortar with dust flying.
3) Cut off the double-ridge bead from the new tile, the part that has to slip under the tile to the left. This should enable me to put the new tile in easily, and then I'd run a bead of polyurethane caulking where the missing bead was cut off. Doesn't matter if a little water gets through anyway since this is tile and the roofing underneath will take the water. (You can see the water exit hole at the bottom of the photo, though it's in shadow and hard to see.)