Here's a slightly different approach. I don;t how it works out for efficiency, but it doesn't require any backtracking.

$_ ='the quick "brown fox" jumps over the "lazy dog"';

$in = 0; 
s[  
    (?>(.)) 
    (?{ $in = ~ $in if $1 eq '"'  }) 
    ([\t ]) 
]
[ $in ? $1.$2 : $1 ]gex;

print

thequick"brown fox"jumpsoverthe"lazy dog"
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You didn't specify whether you wanted the quotes removed , so I left them in.