That is not usually powdery mildew, its a wild yeast that florishes on the leaves. This is a somewhat sticky leaf plant and some of that is also house dust. Avoid wetting the leaves, but give it some strong sunlight. Most rosemary types are not very cold hardy, so they can die over winter from killing frosts. Its actually in the evergreen family and if you notice, some pine trees also get a whitish coating on their needles.

This condition will arise if you don't have enough air circulation. Purchase a small fan and keep it on 24/7, but first take your plant to the sink and turn its top into the sink, then wash off all the leaves. Use your fan or a hair dryer to dry off the foliage. Rosemary can become root bound in a pot. Lift you plant out of the pot, and if there are nothing but roots, you'll need to root prune. March is a good time to root prune. Remove your rosemary, use a serrated knife and cut from the sides and bottom. Replant in some new potting soil, and in a couple of weeks fertilize with a good organic fertilizer. Pot bound plants are stressed and suseptible to disease. You can also repot into a larger pot, but you will still need to root prune, so the roots will not continue to wrap around. Watch your watering, because the plant is not growing and overwatering will cause more problems.

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Thank you for the good suggestions. Would it be a good idea to cut off the new growth that's affected? It's only the 6" of sticky leaves that have been put out since taking the plant in last October. The older, stiffer needles are not coated.

Powdery mildew is very common on rosemary brought indoors. The fungus actually penetrates the leaves as it matures and therefore cannot be washed off. You may be able to rub off the white coating on early infections but the fungus is microscopic and will remain behind to infect again. Often, high daytime temperatures followed by low night temps will cause an outbreak (leaves get damp under these conditions). Fluctuations between high and low humidity also will contribute to the problem. Good air circulation certainly a good idea but won't cure it. The time honored home remedy is a mixture of baking soda and water, 1 tablespoon to 1 quart of water. Spray with a hand mister. Depending on the severity of the problem you may see some minor spotting damage left behind when the spores are killed. You may also have to spray more than once. I recently switched to an OMRI approved consumer product called Green Cure which is potassium bicarbonate. It seems to work a little better than baking soda for me. There is also an old home remedy using milk and water, but I have not had success with this method. Apparently it takes repeated treatments to be effective. I have 100's of plants so this type of regimen is not practical. Sulphur spray or oil sprays can also be used but I have found they are ineffective and more often damage the plants or kill them outright if not careful.

Treat the whole plant. The fungal spores are all over the plant, it is just the young tender growth that is most susceptible and affected first.

I have never heard of a "wild yeast" that causes problems on rosemary or plants in general.
While pines and rosemary are both evergreen plants, there is no "evergreen family". Being evergreen is not a characteristic that is used to establish the botanical classification of or relationship between plants. They are about as distantly related botanically as can be. Comparing the two and what ails them is like comparing apples to peanuts.

Okay, we had a warmish sunny day and I took the plant out and pruned it, top and bottom, gave it new dirt in a new pot and set it out to dry well. I'll try the baking soda and water too, but hope the initial efforts will help. Thanks!