I am cooking Boston Butt at 165 degrees;
after an hour, the Anova had heated the water temperature to 165 degrees;
however, after about 3 hours, the temperature readout on the Anova dropped to 112.2 --even though the setting was still at 165 degrees; now readout has gone to 236.8!!
Any similar experiences and thoughts?
The head of the APC is prone to overheating due to the vents at the back of the unit, if you’re cooking in an area that isn’t well ventilated, where the steam from the water bath can impact the electronics in the head unit.
Having a covered vessel eliminates this potential.
Best thing you can do for uncovered vessels is to be in a well ventilated area, make sure the head of the APC is facing out at the back of the unit (ie. the display face of the unit is pointed towards the centre of the vessel).
Hopefully you have a good digital instant read thermometer, so you can determine what the temp of your water actually is (what is showing on the display may be incorrect).
If you have a lot of condensation on the head of the APC, I’d likely shut it off, remove it from the bath, wipe off the head of the unit with a towel, getting it good and dry…lay it on its back so any potential moisture inside may run out…and leave it laying that way for about 20 minutes (your water bath won’t cool very much in 20 minutes). Then put the APC back into its clamp, in the orientation I’ve suggested and resume your cook.
If you like cooking sous vide a lot (I think for most of us, it’s our go-to for all meats) 
you’ll likely want to investigate a covered vessel. Either the polycarbonate ones or making one out of a camping cooler (which the insulation minimizes any heat loss) are your best solutions. Anova also sells a lid for pots (but you could make one yourself with a plastic/rubber lid and a 2 3/8" hole saw)
Whoops…forgot the obvious one - if it actually isn’t steam wreaking havoc on the APC’s logic, then you may have a defective unit - and hopefully you’re still within your 2 year warranty period - use the support option at the top of the page to contact Anova to get a warranty replacement.