What could be the cause of local AM broadcast band signals combining to generate spurious signals on the MF or HF bands?
Why Corroded joints between metals, in fences, gutters, guy wires, tower hardware, rain gutters, behave like crude diodes. A nonlinear junction sitting in the strong field of nearby AM broadcast transmitters mixes those carriers, producing sum, difference, and harmonic products that are then reradiated and heard as spurious signals across MF and HF. This is often called the 'rusty bolt effect,' a form of passive intermodulation, and the cure is to bond or remove the corroded connection, not to change your receiver.
Watch out Receiver front-end or IF overload can also create broadcast mixing products, but that problem exists only inside your radio and is fixed with attenuation or a broadcast-band reject filter; the pool question points at signals actually being generated and radiated outside the station. Overmodulation would splatter near the offending station's own frequency, not produce products on the HF bands.
Rusty bolt = accidental diode mixer. Corrosion plus strong AM carriers equals homemade intermod.