Answer: This is a complicated question!
Some gardeners will allow the tulips to bloom, wait while the foliage ripens, and then lift and store the bulbs. This process is a lot of work, but it allows the gardener to replant the area with annuals without working around the bulbs. Other gardeners will simply lift and throw away the tulip bulbs, treating them as annuals. Still other gardeners will leave the tulips in place and work around them.
The best method will depend to some extent on the type of tulip you have planted and how long they have been in place and what you want to do with that area of the garden -- and how much time and energy you have. Some tulips such as the species tulips perennialize well and do best when allowed to mature and die back naturally. Other tulips seem to last a year or two and then deteriorate and are probably best replaced (or moved to a nursery bed for extra nurturing to try to rebuild their size) no matter what type of care they are given. Still others seem to do equally well/poorly whether lifted or not, as long as the foliage is allowed top mature and ripen before they are moved.
The problem is that tulip bulbs naturally divide and consequently the result is more but smaller bulbs and some simply perform better (or worse) under garden conditions. Smaller bulbs will not produce the same flowering effect as the original large bulbs you planted. In most cases, the blooms will become smaller and eventually nonexistant. So the bulbs may only bloom satisfactorily for a year or two or three before they need to be replaced or possibly become so crowded that they need to be lifted and replanted further apart.
The choice is basically up to you. Lifting or not lifting will nopt prevent the bulbs dividing, but it will allow you to sort them for size.
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