I think you might have gotten this the wrong way: All decidable problems are trivially reducible to the halting problem (just decide them and then map to a halting/non-halting Turing machine). On the other hand, if you can reduce the halting problem to a problem, it means it's undecidable.
I would be curious though which "natural" undecidable problems people claim to solve regularly.
The most common variations are simply obfuscated variations where it is not clear that a sub-part of the process is turing complete, and where that sub-part can determine whether or not a specific state is reached.
Quite a lot of software have subsystems that turn out to be Turing complete, whether through the explicit inclusion of scripting support, or through more convoluted means.