The literature features many instances of spacetimes containing two black holes held apart by a thin distribution of matter (strut or strings) on the axis joining the holes. For all such spacetimes, the Einstein field equations are integrated with an energy-momentum tensor that does not include a contribution from the axial matter; the presence of this matter is inferred instead from the existence of a conical singularity in the spacetime. And for all such spacetimes, the axial matter is characterized by a pressure (or tension) equal to its linear energy density, which are both constant along the length of the strut (or strings); the matter is therefore revealed to have a very specific equation of state. Our purpose with this paper is to show that the axial matter can be introduced at the very start of the exercise, through the specification of a distributional energy-momentum tensor, and that one can choose for it any equation of state. To evade no-go theorems regarding line sources in general relativity, which are too singular to be accommodated by the theory's nonlinearities, we retreat to a perturbative expansion of the gravitational field, using the Schwarzschild metric as a description of the background spacetime. Instead of a second black hole, our prototypical system features a point particle at a fixed position outside the Schwarzschild black hole, attached to a string extending to infinity. While this string prevents the particle from falling toward the black hole, a second string is attached to the black hole to prevent it from falling toward the particle. All this matter is described in terms of a distributional energy-momentum tensor, and we examine different equations of state for the strings. To integrate the field equations we introduce a new "Weyl" gauge for the metric perturbation, which allows us to find closed-form expressions for the gravitational potentials. Our solutions are linearized versions of multihole spacetimes, and some of them feature strings with a varying tension, unequal to the energy density. We describe the properties of these spacetimes, and begin an exploration of their extended thermodynamics.