Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves
to the category of descent data in the derived category of coherent sheaves.
Nonuniqueness of gluing amounts to $H$ not being a full functor
and is observed for covers of cardinality 3 and higher.
Nonexistence of gluing amount to $H$ not being essentially surjective
and is observed for covers of cardinality 4 and higher.
For covers of cardinality 0, 1, and 2, the functor $H$ is an equivalence.
Let's analyze first the case of covers of cardinality 3.
Given three opens $U_1$, $U_2$, $U_3$,
an object in the domain of $H$
will have chain complexes of coherent sheaves $F_i$ over $U_i$,
isomorphisms $t_{i,j}:F_i→F_j$ of their restrictions to $U_i∩U_j$ ($1≤i<j≤3$, setting $t_{j,i}=t_{i,j}^{-1}$),
and a choice of a homotopy $h$ (or rather its homotopy class) from the composition $t_{3,1}t_{2,3}t_{1,2}$ to the identity map on the restriction of $F_1$ to $U_1∩U_2∩U_3$.
In particular, objects with the same data of $F_i$ and $t_{i,j}$, but nonhomotopic $h$ are nonisomorphic.
An object in the codomain of $H$ has a similar description, except that $h$ is merely required to exist, but its data is not included.
It is now clear how to show that $H$ is not full: pick two objects in the domain of $H$ with the same data of $F_i$ and $t_{i,j}$,
but different (nonhomotopic) $h$.
Their images under $H$ will see the data of $h$ discarded, so the two objects become isomorphic in the codomain of $H$,
despite not being isomorphic in the domain of $H$.
Therefore, to construct a concrete example, one needs to specify $U_i$, $F_i$, and $t_{i,j}$
such that the identity map $p$ on $F_1$ restricted to $U_1∩U_2∩U_3$
admits a homotopy $h$ from $p$ to itself
that is not homotopic to the identity homotopy from $p$ to itself.
That is to say, we want $\def\Aut{\mathop{\sf Aut}\nolimits}\def\End{\mathop{\sf End}\nolimits} π_1(\Aut(F_1))$ to be nontrivial.
For example, we can take $\def\cO{{\cal O}} F_1=\cO[0]⊕\cO[1]$, then $$\End(F_1)=\cO[-1]⊕\cO[0]⊕\cO[0]⊕\cO[1],$$ and a nontrivial 1-cocycle in $\Aut(F_1)$ yields such a homotopy $h$.
The case of covers of cardinality 4 is similar.
As before, for an object in the domain of $H$ we have $U_i$ and $F_i$ ($1≤i≤4$), $t_{i,j}$ ($1≤i<j≤4$), $h_{i,j,k}$ ($1≤i<j<k≤4$),
and now on $⋂_i U_i$ there is a homotopy $q$ from a certain composition of homotopies $h$ to the identity homotopy on the identity map on the restriction of $F_1$ to $⋂_i U_i$.
The functor $H$ discards the data of $h$ and $q$.
In the codomain of $H$, some maps $h$ must exist (but their choice is not given to us),
but $q$ need not exist at all.
Indeed, by modifying the above example appropriately, we can construct descent data for the derived category
in which the composition of homotopies $h$ is not homotopic to the identity homotopy,
which means that the existence of gluing is violated for the derived category.
