As an aircraft flies across the globe, it must continuously pitch (with respect to an inertial frame of reference) to keep following the curvature of the Earth.

From that point of view, what must the sign of this continuous (though extremely small) pitch rate be positive or negative?

The sign of the continuous pitch rate must be negative.

In order for an aircraft to follow the curvature of the Earth as it flies across the globe, the sign of the continuous pitch rate would be negative. This means that the aircraft must pitch downwards slightly to compensate for the Earth's curvature. This pitch adjustment is necessary to maintain a constant altitude above the curved surface of the Earth.