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Basic Trick
The gifts (those you get from the balloons) when you visit another city must be opened AFTER you have shopped in that city.
When you open the thought balloons, you often receive gifts.
In your city it is random, you may or may not receive gifts when you open them. However, when you visit another city, if there is a balloon, there is a gift for sure.
On the other hand, gifts you ALWAYS receive, whether you have space in your storage or not.
Therefore, what you should do is always first acquire what you have gone to that city for, and finally open the thought balloon with the gift. This way you won't run out of space.
In the video below, you can see that you first buy in the city, and then open the balloon with the gift.
