Hah, this is IA-64. It has special hardware support for register spills, and you can search for “NaT bits” here:

https://portal.cs.umbc.edu/help/architecture/aig.pdf

to discover at least two magical registers to hold up to 127 spilled registers worth of NaT bits. So they tried.

The NaT bits are truly bizarre and I’m really not convinced they worked well. I’m not sure what happens to bits that don’t fit in those magic registers. And it’s definitely a mistake to have registers where the register’s value cannot be reliably represented in the common in-memory form of the register. x87 FPU’s 80-bit registers that are usually stored in 64-bit words in memory are another example.

Old-time x86 sort-of has “states representable in registers but not main memory”, too.

Compilers used to use its 80-bit floating point registers for 64-bit float computations, but also might spill them to memory as 64-bit float numbers.

https://hal.science/hal-00128124v3/file/floating-point.pdf section 3 has some examples, including one where the assert can fail in:

  int main (void) {
    double x = 0x1p-1022, y = 0x1p100, z;
    do_nothing(&y);
    z = x / y;
    if (z != 0) {
      do_nothing(&z);
      assert(z != 0);
    }
  }

  void do nothing (double *x) { }

I assume they were stored in an out-of-band mask word