Was the off-shell mass of photons measured in the recent Breit-Wheeler experiment? Premise: The Breit–Wheeler process is a pair production process in which a positron–electron pair is created from the collision of two photons.
Background: In relativistic QFT, solutions that obey the mass-energy-momentum relation are called "on-shell" while those that do not are called "off-shell." The former are referred to as "real" and "physical" while the latter are referred to as "virtual." This, naively, seems to imply that virtual or off-shell particles should not be physically measureable (i.e., according to the energy-time uncertainty principle, they appear and disappear in a timescale that is smaller than what we can measure). However, the result of the recent Breit–Wheeler experiment is interpreted as measuring the off-shell photons which have an (effective?) mass, and not the on-shell photons which have zero rest mass.
Questions: Can off-shell mass be measured in principle? Was off-shell mass of photons measured in the recent Breit-Wheeler experiment? If so, how did that (effective?) mass compare to the mass of the gold atoms used in the experiment?
Note: I'm out of my expertise here, feel free to correct me (especially regarding my understanding of the interpretation of the Breit–Wheeler experiment). An answer that is well cited is more likely to be accepted.
 A: Check the ArXiV version of the paper https://arxiv.org/abs/1910.12400, specially the diagram of the process and the following extract.
I quote

In Quantum Electrodynamics (QED), different processes of creating an e+e− pair from the collision of two photons are defined depending on the virtuality of the photons and on whether the consideration of higher-order processes is necessary. There are three possible interactions: the collisions of two virtual photons (as calculated
by Landau and Lifshitz, giving the total cross section for $e^+ e^−$ production predominantly at the pair threshold [10]), of one virtual and one real photon (Bethe-
Heitler process [11]), or of two real photons — the Breit-Wheeler process [4]. It is important to note that all three processes can be identified in particle colliders in specific kinematics [12, 13].

First things first: There is no such thing as "off-shell" mass, any four-momentum that does not satisfy the particle's dispersion relation is simply off-shell, there is no one single value for this.
So the BW process only pertains collisions of two real photons and to my understanding there is no measure of off-shell mass in the experiment.
Then the last question must not be answered.
Also I am afraid they are being too broad with the term virtuality, and perhaps "soft" is more appropriate, at least in the context of the Bethe-Heitler cases and Bremsstrahlung. You can check the other questions in the site talking about (on)off-shellness.
However virtuality in this context can have measurable consequences as it is described in this paragraph

Since photons are spin 1 particles, in general their helicity ($J_z$) may take values $−1,0$, or $+1$. While real photons are massless and do not allow the $J_z = 0$ state, short-lived virtual photons may carry a virtual mass (virtuality) with a possible $J_z = 0$ state in their role as an intermediate propagator of the electromagnetic force. The consequences for the produced e+e− in a collision of two real photons are a dramatic suppression of the production of vector mesons (spin 1 particles) and a preferential alignment of the $e^\pm$ momentum along the photon propagation axis (i. e., an anisotropic distribution in the polar angle $\theta$)

A: 
Can off-shell mass be measured in principle?

From the kinematics of each event, in the case of off mass shell gamma gamma collisions, one can calculate the four vector sum  of the two  gamma, but each event will have a different value, many events summed giving the calculated distributions. Off mass means that the mass is not zero but varies according to the available limits.
See here a simple example of virtuality


Was off-shell mass of photons measured in the recent Breit-Wheeler experiment?

No. The gamma four vectors ( the "length of the four vector is the mass) involved in the calculations for the experiment cannot be separately computed for each event.

If so, how did that (effective?) mass compare to the mass of the gold atoms used in the experiment?

Like apples with oranges. The gold atoms are the source of the electromagnetic waves, the gamma are the photons building up the electromagnetic waves.
