"Sludge" at the bottom of matcha/green tea latte When I get a green tea latte from Starbucks, the drink is uniformly mixed from top to bottom, no matter how long it takes me to drink it.
I prefer to make my matcha lattes at home, though, because its faster, cheaper, and better (both health and taste, as I prefer my lattes far less sweet than Starbucks makes them and I prefer to sweeten with honey) than running through the drive through every morning.  However, every morning, no matter how fast I drink my morning beverage, I wind up with a layer of green sludge at the bottom of my cup.
From what I can tell, the matcha is falling out of solution with the rest of the drink, despite being thoroughly mixed to start out.  I do wonder if it has something with the amount of sugar, or the use of water to dissolve the matcha in.  But since the whole thing comes together and tastes wonderful (until the last bit), I'm not sure.  The only way around it I've found so far is to stir the drink well before every sip, which is unnecessary with coffee house versions (as I've had them other places and do not recall winding up with sludge at the end of the drink) and annoying while I'm at work.
Why do my matcha lattes end in green sludge, and how can I fix it?  The green sludge at the end is bitter, thick, and unpleasant to drink, so I want to make my latte more like the ones from a coffee shop.
Recipe and method:

3/4 - 1 cup milk (depending on mug size), heated
1 slightly heaping teaspoon matcha powder
1 - 2 teaspoons honey
~1/8 cup hot water (a minute or two off the boil)
Force matcha through a sieve into mug to remove lumps, add hot water.
Blend (using electric whisk/milk frother) until dissolved.  Add honey,
blend again to dissolve.  Froth milk, then add to mug.  Blend to mix
milk and tea/honey mixture.  Drink.
If you are making your drink with real matcha, that matcha is very finely ground green tea leaves. There are parts of the tea leaves that will always be insoluble in water, and therefore eventually fall to the bottom and create this sludge to which you are referring. The only solutions to that issue would be either 1) filtering the solution to remove the insoluble solids or 2) allowing the insoluble solids to settle to the bottom and the decanting the beverage above that sludge into another container. Straining is not enough when you are dealing with very fine particles, as the openings in strainers would just allow those particles to pass right through.

If, however, you are not using real matcha and it is completely water soluble (maybe a freeze-dried preparation from brewed tea) there will be no sludge. I would suspect that Starbucks is not brewing green tea with real matcha for every serving.