If you view the bucket and the water as a single object, a "bucket-o-water", it's centrifugal force that makes the bucket want to fly out of your hand, and your finger strength provides centripetal force, which retains the bucket.
The forces aren't phantom, by any means. They follow Newton3, and are equal and opposite in their application. They also don't just occur. The application of either force depends wholly on the frame of reference from which it is observed.
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"¿ʞɐǝɹɟ ɐ ǝɹ,noʎ uǝɥʍ 'ʇı ʇ,usı 'ʎlǝuol s,ʇı"
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