Letting the compound and the polish "dry" may be your mistake. It looks like polish residue that was not removed OR something from the applicator pad that was not washed or cleaned from it and you have previously used that applicator pad for applying a wax, and then used it for applying the compound and then the polish. I am guessing.
You could rewash the clouded/grey area and then try a smaller area , like a 18 by 18" square, first with a hand application of the compound and polish using two NEW foam applicators, one for the compound and one for the polish and view the results for yourself. Make sure you wipe it off immediately after your application. I doubt you have gone down to the primer using a compound and polish by hand, unless the base coat and clear coat substrate that are on top of the primer were "seriously" corrected (AKA, thinned) with a machine buffer before that you do not know about. Without a Paint Thickness Gauge (PTG) measurement to verify this, there is no way of knowing. Like you mentioned, I would not suggest doing this in the direct sun if it is hot, but either inside a garage or early in the morning or later afternoon when it is cooler and you could use the shade from a building if doing this outside. If the first test area looks satisfactory to you, then move on to the next small area and repeat the process. Just make sure to clean out the applicator pads after doing three or four small areas so you don`t overload the pad with compound/polish product OR if the pad looks dirty. You can use Dawn Dish Soap and HOT water if that is all you have for cleaning, but an automotive all-purpose cleaner and HOT water would be better. Just make sure to wring it out and wrist-snap it to try to remove as much water as possible before reusing it again.
If that FIRST small test area still looks terrible or even worse, you have much bigger "problems" that will require the services of a professional detailer or body shop. I say that because of the limited products you have (Meg`s Compound and Polish, and I assume they are the "Ultimate" consumer versions and not the Professional Mirror Glaze M105 and M205 or the newer Ultra-Pro M110 and M210) and the fact you are doing this by hand.
Also, the microfiber clothes you are using could make-or-break the end results. Some microfibers will lint like crazy and continuous wiping with them to try to remove that lint may mar or "scratch" the surface. Some cheaper microfibers tend to scratch the clear coat as well and others may not be absorbent enough to remove the compound or polish. AND, as with the applicators, if you re-use a microfiber clothe/towel that may not be clean and had been used previously for wiping off some other car-care product, like an interior dressing, you may be getting contamination/streaks from that. Again, I am guessing.
Sorry if I sound like a put-down and am so critical, but some detailing "problems" are self-inflicted simply because of ignorance and not knowing proper application and removal techniques and methodologies and having the right accessories/equipment for obtaining the desires results.
Just like removing a frozen hex-head bolt with an adjustable wrench and hammer may get the job done, but chances are you`ll round off the corners/flats, when using a penetrating oil first and then a hex socket and breaker-bar may have been a much better choice, but you have to have the latter two product and tools to do that to begin with. If all you have is the adjustable wrench and hammer, you use what you have and may suffer the consequences.