Hi Elliot,
A couple quick thoughts to share.
Agitation and FiltrationAgitation is key. If you can add it to your type of sprayer, it will definitely help you with almost everything you would spray on your trees, year round.
I dont think more soap is going to help with the particles you are ending up with in your mixture. If after adding liquid soap, you see oil still floating on the top, then go ahead an add another splash of liquid soap and mix again. Still see oil coming to the surface? add a little more soap and repeat. You need only add enough soap to emulsify the neem and more is not going to magically help the particles that are not soluble to dissolve. According to Usha at Neem Resource the solids that are "thrown" from the neem oil are
stearin . . . Stearin needs to be filtered out before adding it to your tank. Stearin is a wax like fatty acid substance that has a high melting point of greater than 130 degrees Fahrenheit - which you are not going to achieve without practically cooking your neem oil . . . and you don't want to do that for numerous reasons.
I am careful to always strain my Neem mixture (after emulsifying it) as I pour it into the tank. Even with an excellent emulsion, there are always a few bits and pieces that do not dissolve completely - any of which can easily clog a sprayer. These "bits" may be the coagulation you are talking of, that is, they are actually many hundreds of particles that never emulsified during your mixing and are seemingly falling into a group, at the bottom of your tank, thanks to gravity and then appear like a precipitate coming out of the solution. . . . try straining your mixture more thoroughly before pouring it into your tank, I am confident that will help.
I have been using a Rears Nifty series 50 gallon tank with agitation, over the last 3 seasons, and I have not had Neem cause any clogging in my tank hardware or sprayer since switching to that product.
I also have moderately hard well water that I use for my tank mixing. I test my well water every 2-3 years and a recent lab test result from Sept 2015 was 151mg/L of CaCO3 and a pH of 7, if that helps with your analysis.
Good luck!
Gopher Hill ApplesZone 8 in California