I don't have too much fusarium or verticillium but I do get mostly late blight and mildew since I live in a hot climate. Sanitation is important as well as selecting suitable varieties that are disease resistant. It does mean that I often have to settle for less than spectacular tomatoes but they still taste better than the market varieties. The spores of some fungal diseases are floating around in the air and hard to totally avoid.
However, the conditions for fungal diseases to take hold are the same.... warm humid conditions.
You will rarely see fungus on the plants when it rains almost continuously for weeks but within three days after the rain stops and the humidity increases from all the water evaporating off from the ground, that is when fungal conditions take hold. It is important to start an antifungal spray program as soon as the rain stops and to repeat it if you have light rains between. While it is raining, the rain washes off the fungal spores before they can take hold, it is when the humidity rises and the spores are still sitting on the leaves, when the plants are vulnerable. Sometimes no matter what I do, I can't keep the fungi off and I end up pulling the plants, waiting for dryer weather and starting over.
Plants that are most susceptible will be plants that come from dryer areas and have leaves that are hairy or fuzzy. These plants are designed this way to actually capture moisture, but it is a problem when they are in a wetter environment. Sometimes, if you can wait, it is better to move the plants out when the weather is dry. The alternative would be to actually keep the plants wet and keep washing off the spores, but I haven't been able to keep them wet long enough to get past the humidity without having issues form the soil being too wet.
Sometimes, I have had to rotate fungicides. It is important when using sulfur or oil not to use them within a month of each other or they will burn the plants. If I know the rain is going to last awhile, I usually will go with an oil as a preventive since it helps the plant shed water and does not wash off like sulfur. If it is really humid, I might add a second fungicide like chlorothalonil or even cinnamon, potassium bicarbonate + horticultural oil, or milk.
https://pdfs.semanticscholar.org/8140/3 ... eed8f5.pdf
Happy gardening in Hawaii. Gardens are where people grow.