Building permits on Ambergris Caye: What are the requirements for building on Ambergris Caye? Permits, time constraints, can we be our own builder and can we bring our own plans from a USA certified architect?

You will need a building permit.  Complete plans are to be submitted to the local housing authority for review, after which permits are issued.
 
Go to the Town Board and ask for zoning and other conditions that affect your area before designing a home.    The plans you submit must be drafted by a local, approved architect.
 
You can legally be your own builder, but I don't recommend doing this yourself.    The learning curve is expensive, and there is no way to tell you all you need to know in advance.  
 
Supplies are a whole other subject.  They come from here and there, and learning sourcing takes time.  Lots of building goes on here so materials are available but learning logistics is not so easy. 
 
If you do hire a crew on your own, get copies of labour laws and be sure to follow them carefully.
 

Building plans must be stamped by a Belizean architect, along with PUC, etc.

You cannot just bring your plans from elsewhere and run with them, although you could bring preliminary sketches and have a local architect convert them to actual build able plans.
There are many requirements, which should be discussed with a local architect. The will include particulars related to foundation, septic, etc.

You can only work in Belize if you have a work permit or are a legal resident. An owner-builder (even just in a casual-supervisory capacity) would need a TaxID number to employ workers, etc - this is not just a “go do it” situation.

The largest building you can build in Belize without a permit depends on the municipal jurisdiction.

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