I have a difficult time believing that in day to day driving, that the automatic beats the manual. BUT, if it does, it's worth knowing.
At least in older generations, the 5MT Corolla will beat the 4AT mileage in city driving (if that is what you refer as day-to-day driving) because upon acceleration there is very little losses on the MT, except at the very start when the clutch is slipping. In the automatic version, much of the power is lost through torque converter in city driving.
In highway driving it is a different situation, because at those speeds the torque converter goes into lock-up and power is transferred from engine to wheels in basically the same way as it is done on MT.
What I have found in general on Corollas is that the 5SPD MT is geared with close-ratio between gears, providing with pretty good acceleration and it makes the MT model much more fun to drive than the AT which seems to lack some juice...But that doesn't mean that the 5th speed on the MT is more fuel-efficient than the overdrive of the AT.
I can only compare with 9th gen MT and 8th gen AT, but at 100 km/h the MT is geared in such a way that the engine revs at 2600rpm exactly at 100km/h. The 8th generation AT, on the other hand, revs MUCH lower than that. That might explain some fuel efficiency differences on the highway.
Actually I would be interested to find out from someone here what their RPM is displaying at 100km/h in lock-up situation.