There is more to this question than meets the eye. The question is related to the question 'Gravitational' power vs. engine power and this answer will be related to the answer https://aviation.stackexchange.com/a/56040/34686 . That question used the phrase "exactly identical" and so a very precise answer was given. At the end of the current answer we'll look at whether or not these considerations make any PRACTICAL difference for flight in conventional general aviation aircraft.
You say both aircraft are on the same glide path-- meaning, no doubt, relative to ground. This means the aircraft with a tailwind is following a steeper glide path relative to the airmass within which it is flying. This means that the glide path through the airmass is aimed more steeply down, relative to the horizon, in the case of the aircraft with a tailwind. This means that the aircraft with a tailwind must be flying at a lower power setting than the aircraft with a headwind.
Due to the relationships explained (and now also illustrated) here https://aviation.stackexchange.com/a/56040/34686 , the lift vector is slightly SMALLER in the case where the lift vector is tilted forward more, i.e. where the flight path through the airmass is aimed more steeply downward. This describes the aircraft flying at the lower power setting-- the aircraft with the tailwind.
If angle-of-attack were the same in both cases, the aircraft with the tailwind would be flying at a slightly lower airspeed-- that is the only way it could be producing less lift, for the same angle-of-attack.
Since you specify that airspeed is constant, the aircraft with the tailwind must be flying at a slightly lower angle-of-attack than the aircraft with the headwind. (This introduces the complication that the L/D ratio will not be exactly the same in both cases-- but the basic ideas explored in the answer https://aviation.stackexchange.com/a/56040/34686 will still apply.)
It is very unlikely that a pilot would be able to detect this difference in actual practice in most cases. In actual practice, the angle-of-attack would probably seem to be the same in both cases. For example, consider an aircraft with a L/D ratio of 8:1 at idle power at some given angle-of-attack. In still air, the aircraft can glide 8 feet forward from every foot of altitude it loses. The glide angle at this angle-of-attack is equal to arctan (1/8) = 7.1 degrees, and the lift vector is equal to weight * cosine (glide angle) = weight * cosine (arctan (1/8)). This works out to weight * .9923. So there is a less than 1% change in the magnitude of the lift vector between the idle-power case, and the case where aircraft is generating enough power to fly horizontally at that same angle-of-attack. This means the airspeed need only change by a factor of (square root of (.9925)). or .9961 .
If we are talking about an aircraft with the glide angle of a brick-- like an aircraft with a L/D ratio of 3:1 or 2:1 or less -- then the difference in the power-on and power-off cases would be more significant, regardless of whether it is the airspeed or the angle-of-attack that we are constraining to stay constant.
So the PRACTICAL answer to your question is, that for most things that fit our definition of an "aircraft", there is essentially no difference in angle-of-attack required to fly down a given glide path (defined relative to the ground) with a tailwind or headwind, with the same airspeed in both cases.
By the way, anyone unclear on more basic concepts such as the idea that even in the presence of a strong wind, no changes in angle-of-attack or power setting are needed to hold a constant altitude and airspeed while "circling" at a constant bank angle (and here we mean that the flight path through the airmass, not the ground track, is a round circle) should probably focus on those basic concepts rather than the nuances explored in this answer.
See related answers to related questions:
"'Gravitational' power vs. engine power" -- https://aviation.stackexchange.com/a/56040/34686
"Does lift equal weight in a climb?" --
"What produces Thrust along the line of flight in a glider?" -- https://aviation.stackexchange.com/a/56371/34686