Foreign Employment for H1B H Visa

I understand H1B visa holders must be sponsored by their employers.
What if, while legally employed in the US, an unrelated company located in Europe offers consulting work to be done by email?

Physical location is irrelevant; the work is pure analysis and would be sent back to Europe to be used there, not in the US. In fact, the work could conceivably be done while on a beach in the Bahamas.

Can the H1B visa holder do the work and treat the money received as foreign income, or is it more complicated?