2x10 joists set 16" o.c. is as basic as it gets. If they span close to 13', I'd say you're pushing the limits for ceramic tiles, plus we don't know the species or grade yet. So I'sd say your joists are marginal at best, but the joists deflection isn't the most important, the deflection between the joists is more critical, which is covered by the subfloor, the ⅝" ply.
⅝" ply subfloor is pretty wimpy, although it does meet minimum standards if it's installed correctly and in perfect condition. What are the chances, and do you wanna chance it?
If you can check the joists info and get good numbers it would be helpful. Is the floor below this the first floor or a basement? It's possible to measure the joists to confirm they're 2x10" and how far they span without seeing them. Just gotta do some detective work. The species and grade might not be so easy.
So, I'm not sure about the joists right now, but I think you should install min. ⅜" underlayment grade ply plus Ditra to save height, or go with thicker ply if you can.
TILE GUY - retired- TROY, MI - Method & Product suitability consulting.
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