Take, for example, the shape of new radio antennas in cell phones. Instead of a linear arrangement, changing the shape to a fractal design in 4 dimensions improves cell phone reception dramatically -- to the point that instead of requiring a 6 inch long pole, a die smaller in area than your pinkie fingernail produces multiple-times improved reception.
There you have empirical evidence of a dimension beyond our own which influences and is part of our own. So the question, then, is how you split the difference between a "supernatural" realm and an "extranatural" realm. Unload the term "supernatural" of its cultural connotations (as you suggest expatrie's argument relies on them) and take it at face value to mean "above natural" -- categorically, is there a difference? Is there, or is there not, empirical evidence in support of a realm above/beyond our own?