Symbolic references, which the initial question is about, require entries in the symbol table. "my" does not place entries in the symbol table. This explains the behavior the initial poster asked about, and also answers why you will get

b = 1
b = 2
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for your output. The setting of $$a to 2 sets the global $b to 2, but inside the block $b refers to the lexical $b with no symbol table entry.
Hope this helps...
Other posters have mentioned that what the original poster may really want to use are the hard references...

-JAS