MD-11 Captain reported fuel system anomalies shortly after reaching cruise altitude. Crew diverted as fuel indications were suspect and aircraft had just returned from heavy maintenance event.

Narrative

Shortly after reaching top of climb received Level 1 TNK 1 Tip Fuel Lo alert; followed within a minute by TNK 3 Tip Fuel Lo alert. Ran QRH; no help. While troubleshooting these oddly concurrent alerts (and discussing that our jet had recently completed several days of maintenance) and coordinating with Dispatch and Maintenance Control via ACARS; we discovered that we were now showing approx 6000 lbs low to flight plan; less than 20 minutes after we'd noted 2000 lbs positive at TOC. Maintenance Control sent an ACARS that stated that their computers were telling them our jet was currently faulting tanks 1 and 3 fuel floats. As a crew we began to discuss the issue (summary of the situation; recommendations based on what we knew at that time; and comfort levels) - and whether it was it possibly only an indication error; or if our fuel numbers might possibly be grossly in error; even prior to refueling? Also; if necessary; what our divert alternatives and preferences might be? After a period of a few minutes we seemed to recover" a few thousand lbs of fuel; though we were still inexplicably low on gas relative to our flight plan and earlier readings (between 3 and 4 thousand pounds low to plan as I recall). We called Dispatch via SATCOM to discuss these issues and divert options if that became necessary. Moments later we received a Level 2 Fuel QTY/USED CHK alert - a primary alert for possible fuel leak; and initially ran that checklist up to the point of isolating the fuel tanks to determine the probable source of any leaks (none immediately apparent - fuel state didn't worsen relative to plan; although it was still a few thousand pounds low); after which we again SATCOMed Dispatch; asking for Maintenance Control to join; and requesting our Chief Pilot as well. As a flight crew we suspected that it was a gauging error; but also had no way of knowing with confidence whether our ramp load of fuel was also grossly inaccurate (based on previous block-in fuel - our gauge were clearly suspect). While discussing with Dispatch and Maintenance Control; one of the Maintenance Control representatives on the call informed us that their system was presently indicating our jet had between 6 and 7000 lbs of fuel remaining in each main tank (at this point was also showed approx 10;000 indicating in the aux after being transferred forward from the tail tank). If Maintenance Control's information was correct we'd possibly/probably have flamed out prior to reaching ZZZ. Coupled with the previous strange fuel indications and alerts; and considering that the jet had recently completed several days of intensive maintenance; our decision to divert was clear at this point. Because Dispatch was on the call when we learned of Maintenance Control's fuel indications; we were able to rapidly coordinate with our dispatcher the optimal and nearest divert at that point (ZZZ1). We disconnected the call with a plan; contacted ATC; summarized our situation; obtained new clearance direct ZZZ1; and [requested priority handling]. Landed ZZZ1 safely; indications on final suggested weight approximately correct; although initial drip stick readings by maintenance suggest gauging error in #2 tank by between 2300 and 4000 lbs low. Float faults in tank 1 and 3 still unexplained."