We use kinetic particle-in-cell and MHD simulations supported by an observational data set to investigate magnetic reconnection in clusters of null points in space plasma. The magnetic configuration under investigation is driven by fast adiabatic flux rope compression that dissipates almost half of the initial magnetic field energy. In this phase powerful currents are excited producing secondary instabilities, and the system is brought into a state of "intermittent turbulence" within a few ion gyro-periods. Reconnection events are distributed all over the simulation domain and energy dissipation is rather volume-filling. Numerous spiral null points interconnected via their spines form null lines embedded into magnetic flux ropes; null point pairs demonstrate the signatures of torsional spine reconnection. However, energy dissipation mainly happens in the shear layers formed by adjacent flux ropes with oppositely directed currents. In these regions radial null pairs are spontaneously emerging and vanishing, associated with electron streams and small-scale current sheets. The number of spiral nulls in the simulation outweighs the number of radial nulls by a factor of 5-10, in accordance with Cluster observations in the Earth's magnetosheath. Twisted magnetic fields with embedded spiral null points might indicate the regions of major energy dissipation for future space missions such as the Magnetospheric Multiscale Mission. © 2015. The American Astronomical Society. All rights reserved.