For those wondering about drop D tuning, it really shouldn't pose a big issue in this piece.
Tune your string down to D then give it a few decent tugs at around the 12th fret, then re-tune. Repeat. Usually I find myself subtly retuning during the first section if needed. There are a few easy spots. For example at the first sign just before the rall (end of bar 11) there is a good opportunity to give your 6th string a quick tweak seeing as the right hand is occupied with open strings (so the left hand is free). You can do this the first and second time you get there.
If tuning is bad there is an opportunity just before the pui mosso to adjust tuning, but I avoid this usually, it takes away impetus from the second section. But, that is preferable to playing the whole pui mosso with a nasty 6th string.
Btw, tuning your guitar while playing should be a pretty normal thing for you to get used to doing. In this instance you already know that your 6th string is going to be sharp, so it's not as if you need to work out why your guitar sounds terrible.
You will find that with the classical guitar there is no such thing as "standard", nor is there universally any such thing as "perfect" tuning. Actually you want your guitar to be slightly out of tune to add a special richness to the chords associated with this piece.
The most important things regarding tuning for this piece ----> You want to check G on the 6th string, G on the 4th string, and open G on 3rd string to make sure they are in tune. You then want to balance this by making sure both D major chords are in tune and rich sounding. (The one with F# on 4th string and the one with F# on the 1st string). Having a nice rich slightly flattened F# on 4th string is obviously going to counter your efforts to keep your G's in tune (imortant for the pui mosso) but if you can find an acceptable balance between these two, it means your guitar is going to sound well tuned throughout this piece. A final thing to check is your 5th string D with your 4th string open D. This is actually less important than the previous two things, but the second pui mosso can sound awkward if your 4th string is overly flattened to produce that nice F# - but flatten the 5th string also to help you bridge this compromise.
Sorry if that was overkill but I get pretty excited about tuning and stuff.