What reference clock is an atomic clock measured against? I looked at a few of the other posts regarding the accuracy of atomic clocks, but I was not able to derive the answer to my question myself.
I've seen it stated that atomic clocks are accurate on the order of $10^{-16}$ seconds per second. However, if there is no absolute reference frame with which to measure "real-time”, what is the reference clock relative to which the pace of an atomic clock can be measured?
Is the accuracy of an atomic clock even meaningful? Can't we just say the atomic clocks are perfectly accurate and use them as the reference for everything else?
 A: BIPM and TAI
The International Bureau of Weights and Measures (BIPM) in France computes a weighted average of the master clocks from 50 countries. That weighted average then gives International Atomic Time (TAI), which forms the basis of the other international times (e.g., UTC, which differs from TAI by the number of leap seconds that have been inserted, currently 37).
There isn't, however, a single source that gives TAI in real time. Rather, BIPM basically collects statistics from each national lab, computes a worldwide average, and publishes a monthly circular showing how each differed from the average over the course of the previous month. The national labs then use this data to adjust their clocks so they all stay in tight synchronization.
Most of the statistics are collected by using GPS for dissemination. That is, a laboratory will periodically compare their local time to the time they receive via GPS, and send the difference they observed to BIPM. A few links (8, as of the current circular) use two-way transmission of their current time and frequency instead.
BIPM also publishes a weekly "rapid UTC" report with similar information to give national labs slightly more up to date information to help stay in sync better.
To assist the GPS based comparisons, BIPM periodically (most recently in late 2018) does trips around the world to the various national labs with a couple of GPS receivers that are used to calibrate the receivers at each lab.
Individual Labs
The master clocks from those countries are themselves an average of a number of atomic clocks, all stored in vaults to keep them in the most constant environment possible.
These are not, however, all identically constructed though. Let me give the US Naval Observatory's master clock as one example:

The atomic clock timescale of the Observatory is based on an ensemble of cesium-beam frequency standards, hydrogen masers, and rubidium fountains. Frequency data from this ensemble are used to steer the frequency of another such maser, forming our designated Master Clock (MC), until its time equals the average of the ensemble, thereby providing the physical realization of this "paper timescale."


Specifically, the frequency of a device called an Auxiliary Output Generator is periodically adjusted so as to keep the time of this maser synchronized as closely as possible with that of the computed mean timescale USNO timescale UTC (USNO), which in turn adjusted to be close to the predicted UTC. The unsteered internal reference timescale is designated as A.1, while the reference of the actual Master Clock is called UTC (USNO).


UTC (USNO) is usually kept within 10 nanoseconds of UTC. An estimate of the slowly changing difference UTC - UTC (USNO) is computed daily.

GPS
The most easily available reference clock for many people is a GPS signal, so it's probably worth mentioning a bit about it. Each GPS satellite has at least one atomic clock on board (and most have two). These are (occasionally) adjusted by a ground station (Schriever Air Force Base, Colorado), ultimately based on the master clock from the US Naval Observatory.
Also note, however, that most typical GPS receivers will use time from other satellite systems (e.g., GLONASS) interchangeably with actual GPS satellites. In fact, at any given time it's pretty routine that you're using signals from some some satellites from each system. From the user's viewpoint, the two are identical, but GLONASS is a Russian system so (unsurprisingly) it's controlled from a Russian base station and they use their own master clock as the basis for its time, though the US and Russia both contribute to TAI, so the clocks remain tightly synchronized.
Another mildly interesting point: the clocks on GPS satellites have to be adjusted due to relativistic effects--both special and general relativity affect the time (i.e., they're affected both by the fact that they're moving fast, and the fact that they're at high enough altitude that they're much less affected by the earth's gravity than ground-based clocks).
As noted in the section on BIPM and TAI, the various laboratories themselves also use GPS (and GLONASS) for their internal comparisons to help them stay in sync with each other.
Summary
The international standard is based on a weighted average of the standards from 50 different countries, each of which is (in turn) based on a weighted average of a number of separate clocks. The individual clocks are of at least three distinct types (cesium, hydrogen and rubidium).
At least for the US Naval Observatory, the official final output is actually via a hydrogen maser, which is occasionally adjusted to synchronize its current time/frequency with that of the rest of the ensemble.
The unofficial final output used by most people is GPS (or equivalently, GLONASS, etc.) These also include their own atomic clocks, but those are adjusted to maintain synchronization with the ground-based reference clocks.
TAI is approximates the SI second about as closely as current technology supports (and will probably be updated when technology improves substantially--though such a substantial change may easily lead to a change in the SI definition of the second as well). Although it's based on measurements, TAI is never really current--it's based on collecting data, averaging it, and then (after the fact) publishing information about how each laboratory's master clock differed from the weighted average of all the clocks.
References
BIPM
USNO Master Clock
USNO Time Scale
2018 group 1 calibration trip
Explanatory Supplement to BIPM Circular T
A: 
However, if there is no absolute reference frame to measure "real time" for, what is the reference clock that an atomic clock can be measured against?

They are measured against an ensemble of other identically constructed atomic clocks (all at rest with respect to each other and under identical operating conditions). The $10^{-16}$ means that two such clocks will on average drift apart from each other at a rate on the order of a picosecond every few hours.
A: 
what is the reference [...] to measure "real time" [ duration ]

A widely studied general reference for comparing durations is provided within (or: by) the theory of relativity: in terms of (ratios of) arc lengths of path segments of clocks, as each proceeds through a sequence of events "on its timelike path through spacetime".
The duration $\tau[ \, \mathcal A_J, \mathcal A_Q \, ]$ of a material point (participant) $A$, from its indication $A_J$ of having taken part in event $\varepsilon_{AJ}$ (i.e. in coincidence with some other suitable participant $J$), until its indication $A_Q$ of having taken part in event $\varepsilon_{AQ}$ (i.e. in coincidence with some other suitable participant $Q$), is accordingly defined as
$$\tau[ \, \mathcal A_J, \mathcal A_Q \, ] := \text{Infimum} \! \left[ \, \left\{ \, \left( \sum_{k = 0}^n \ell[ \, \mathcal A_{(k)}, \mathcal A_{(k + 1)} \, ] \right) \text{with } n \in \mathbb N, \, \mathcal A_{(0)} \equiv  \mathcal A_J, \, \mathcal A_{(n)} \equiv  \mathcal A_Q \, \right\} \,  \right]$$
where $A$'s indications $\mathcal A_{(k)}$ are of its participation in events of its path segment between  event $\varepsilon_{AJ}$ (at the beginning) and  event $\varepsilon_{AQ}$ (at the conclusion), and the $\ell$ terms represent values of the so-called Lorentzian distance between the respective pairs of events in which $A$ took part.
Note that the infimum is to be evaluated of all sums (as opposed to evaluating the supremum when determining arc lengths of spatial path segements) because Lorentzian distances are superadditive by definition.
Those required values $\ell$, or more correctly: at least ratios of those values, can in turn be measured (definitively) by suitably chosen ideal clocks, such as the geometrodynamic clocks proposed by Marzke and Wheeler.

Is the accuracy of an atomic clock even meaningful?

With the described reference it could determined (at least in principle)

*

*whether a given clock (and especially any given ticking clock, such as an atomic clock) has constant (tick) rate, or how "its rate" (compared wrt. to different pairs of tick indications) varied in suitably extended trials, and


*whether the separately constant (tick) rates of any two given clocks were equal, or by how much they differed from each other.
But: Is this reference actually used in practice ? ...
Apparently not -- clearly it would be awfully cumbersome, laborious, costly, time-consuming and utterly impractical.
However, without using such a rigorous reference, it seems indeed questionable whether we could strictly speak of accuracy of generic clocks at all;
especially considering the possibility of perturbations of unknown "sources or reasons", which moreover might not diminish the (mutual) precision of an actually given set of clocks.
