Two summation methods $\Sigma_1, \Sigma_2 : (\mathbb{N} \rightarrow \mathbb{C}) \rightharpoonup \mathbb{C}$ are consistent iff $\Sigma_1 \cup \Sigma_2$ is functional (right-unique), i.e.
$$ \forall x \in (\operatorname{dom} \Sigma_1 \cap \operatorname{dom} \Sigma_2) : \Sigma_1(x) = \Sigma_2(x) $$
Many of the well-known summation methods (Cesàro summation, Abel summation, Borel summation, Euler summation, etc.) turn out to be consistent with each other. Are there any examples of mutually inconsistent summation methods that are not ad hoc, i.e. motivated by or constructed for the purpose of being mutually inconsistent? If not, is there some explanation behind this fact? Is it possible there's some kind of ideal "general summation" that all these methods are approaching?
Note that this is different from the question of non-constructive extensions such as those given by the Hahn-Banach theorem.
I can think about two examples.
Analytic continuation. This very often gives different results depending on what function we chose to continue analytically.
Ramanujan's summation. There are several slightly different methods under this name, one method for instance, when compared to zeta regularization, removes the pole at $\zeta(1)$. Thus, it gives the correct result $\gamma$ for harmonic series, but around the pole its values differ from zeta function.
Also, technically Ramanujan's summation depends on the derivatives of the function to be summed up, and as such, on the values at non-integer points. But this problem can be avoided by restricting the method only to Newton-analytic functions (those which are equal to their Newton series expansion). This kind of functions, basically, has no arbitrary variations between points, their values at non-integer points are fully determined by the values at integers.