The current 2x6 with 16 inch centers would be strong enough, but I believe you would be unhappy with the deflection (bounce) in the floor.
If it were me, I would sister 2x8's to the 2x6's for the added deflection strength. Especially if you are goung to put ceramic tile down as well.
Another beam supported by posts would also be good, but I would think more work than sistering.
Good Luck!
Dodgeman
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