You can compost using many types of systems. I bought and used a tumbler for two seasons before giving it up for mesh bins. I find the open type mesh bins to be cheap, easy to put up, hold lots of compost, and are easy to use; with either hot or cold composting. Also better in the winter.

It sounds like you may have two problems: perhaps too much moisture in your ingredients, and perhaps the materials are not chopped up finely enough.

One of the things I disliked about the tumbler was the tendency toward clumping. Balls of wet material often formed in the tumbler.

I am not suggesting you cannot make perfectly good compost with a tumbler. Rather, that I found a method I preferred more.

The mushrooms are fine. It got too wet in the tumbler and the fungi bloomed. Dry it out and the flies will lessen as well. Composting is a never ending science project!

Fungi are an integral and even necessary part of the decomposition cycle. Don't be afraid of mushrooms, they are simply the fruit of the fungal networks that are busy breaking down your pile. They in turn will be decomposed by bacteria and other microbes.

If you have flies, it's either too wet, or you have to much nitrogen, or both. Try adding more shredded paper to dry things out, and if it ever stops raining, leave the door open to help some moisture to evaporate. Tumblers seem to be prone to getting too wet.

"Tumblers seem to be prone to getting too wet."

As the tumblers are not able to add the moisture to themselves, I suspect there may be other culprit(s).

Lloyd

Haven't you heard about the marauding bands of raccoons with watering cans who sabotage compost tumblers? :-p

Seriously though, in any given case it's usually not clear whether the excessive moisture is due to the user adding too much water, or whether it's caused by a combination of wet ingredients and no air holes in the darn things.

One thing's for sure, if it doesn't have a way to lose water, once you goof and get it too wet, it's going to stay soggy a lot longer than an open bin or pile would if it doesn't have a way for water to exit.

Both the flies and mushrooms point to one problem, excess moisture in the mix. The mushrooms are an indication that you have an active fungus colony in your compost, a normal part of the digestion of compost process. Those fungi need a fairly moist environment to produce the seed stalks, the mushrooms, that will spread the colony around. The same thing happens in many lawns when that soil gets an ample supply of water.
The flies are looking for some place to lay eggs and one of the places they look for is decaying organic matter that is quite moist and warm.
Some tumblers have provisions for getting rain inside, while the air holes of others can allow rain inside, and that can cause the material to get too wet. Add enough dry material to absorb the excess moisture.

Thanks for all the feedback, everyone!
I have been staying on top of turning the tumbler every day now, so I'm seeing less flies. They are mostly gone, I think. I also added some dried grass clippings to try to soak up some of the moisture(didn't have any browns readily available). I'm going to try to add some paper today. I've been finding it kind of difficult to find enough browns around the house so that is probably part of the moisture problem. My tumbler does have a drain hole, at least I'm pretty sure I've seen it and when I occasionally add water it does drain out the bottom. I'm not sure how, but I really do think rain got in there because it was way more wet than it usually is. I should probably throw a tarp over it next time.