Stochastic unravelings allow one to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations in the Schrödinger picture, which preserve both Hermiticity and trace. In this work we introduce a general framework that extends piecewise-deterministic unravelings to the Heisenberg picture and arbitrary trace-nonpreserving master equations, requiring only positivity and Hermiticity of the dynamics. Our approach includes, as special cases, unravelings of arbitrary dynamics in the Heisenberg picture, evolutions interpolating between fully Lindblad and non-Hermitian Hamiltonian generators, and equations employed in the derivation of full counting statistics, for which we show it can be used to obtain the moments of the associated probability distribution. The framework is suitable for both trace-decreasing and trace-increasing processes through stochastic disappearance and replication of the stochastic realizations, and it is compatible with different unraveling schemes and with reverse jumps in the non-Markovian regime.