Yes. So long as you have enough session quota to go around, you can install the same script on as many sites as you'd like.
With this approach, all of your session data will be intermingled in one FullStory account. You can use search and segments to keep everything straight.
If you'd prefer to automatically silo the data, you'll want additional FullStory accounts, each with its own unique script. Switching between accounts is easy and can be done from within the app. Use the Contact Us button below if you need help adding more accounts.
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