While pointlike localized free fields can be constructed for the positive mass and massless helicity representations of the Poincaré group, just string-localized fields are known to exist for the infinite spin case, the only remaining class of positive-energy representations. [Mund, Schroer, Yngvason 2006] There is even a No-Go Theorem on the existence of Wightman fields with infinite spin. [Yngvason 1970] However, two-particle wavefunctions with compact modular localization [Brunetti, Guido, Longo 2002] can in fact be constructed. In my talk I want to discuss results of my PhD project (Supervisor: Jakob Yngvason) regarding the construction of fields which create these wavefunctions from the vacuum: It turns out that they are incompatible with relative locality to the known string-localized fields due to the conflicting requirements of covariance and momentum-space analyticity. In addition, these incompatibilities are illustrated by a geometric construction which presents the string-localized fields as limits of their massive counterparts.