Should I reinforce the floor over my basment to tile bathrooms?
Floor joists can bend (deflect) without breaking. However, certain kinds of flooring cannot deflect much or they’ll crack (or the grout joints between them will crack).

Each material is rated for maximum deflection. Ceramic tile happens to be one of those materials that cannot deflect much without cracking. 

At a 12’-0” span, most lumber species will deflect too much. However, here are a few things to consider:

1) at 11’-0” most lumber species with a “standard” grade (No. 2 &amp; better) will work. 

2) span distance is measured from face of support to face of support. That is to say, it’s the “clear span”.

3) calculations are an average. That is to say, they use 40 lbs. per square foot for Live Load AND 10 lbs. per square foot for Dead Load. I suspect your live load will be less (take your weight plus your wife’s weight and divide by the room area), but your dead load is greater (weight of tile, mortar base, underlayment, subfloor, joists, ceiling below (if any), etc. )

If your span is 11’-8” or so and you do not weigh 350 lbs. each, I suspect you’re fine.  I’d go look at each joist and see if there are any large knots (especially loose knots). If not, the grade could be No. 1 or select structural and you’re probably ok. 

As an alternative to adding a beam in the basement (happy wife, happy life) you could sister a joist to each of the 5-6 joists in the immediate area.