When you're finished with the soap phase of your project, rinse the tipi well. Then rinse it again.
After you're done with that, rinse it again. And have I told you to rinse it really well? You really must rinse all traces of the soap away, because soap, moisture, and sunlight form a deadly combination for fabric.
Of these three things, the only one you can control is the amount of soap, so for the sake of your tipi's life span, that amount should be zero.
Be sure to let the tipi pieces dry thoroughly before putting them away. Remember that with all this washing and rinsing and rinsing and rinsing, your tent is a lot wetter than it is likely to get even from a downpour.
All-cotton tents may require days of warm weather to dry thoroughly enough to prevent mildew infestation.