Kelly, you have 2 options: 1) add a support beam somewhere close to the middle of the span. 2) add more 2x10s between the ones you have now, so that they will be 8 inches apart. Looking at the pictures, I'm not sure which will be easier for you.
Either way, you need more plywood on the subfloor. Stone is fragile, so it needs twice the support of ceramic tile. It is also susceptible to structural movement, so the second layer of plywood, when installed properly, acts as an isolation layer. 5/8" is not much subfloor for a ceramic tile floor.
So: if you add the beam: add 5/8" (minimum) plywood and your backerboard. The beam should be 2 2x10s with supports every 6 or so feet.
If you add the joists, add 3/8" (minimum) and your backerboard.
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