Coordination of multi-robot systems (MRSs) relies on efficient sensing and reliable communication among the robots. However, the sensors and communication channels of these robots are often vulnerable to cyberattacks and faults, which can disrupt their individual behavior and the overall objective of the MRS. In this work, we present a multi-robot integrity monitoring framework that utilizes inter-robot range measurements to (i) detect the presence of cyberattacks or faults affecting the MRS, (ii) identify the affected robot(s), and (iii) reconstruct the resulting localization error of these robot(s). The proposed iterative algorithm leverages sequential convex programming and alternating direction of multipliers method to enable real-time and distributed implementation. Our approach is validated using numerical simulations and demonstrated using PX4-SiTL in Gazebo on an MRS, where certain agents deviate from their desired position due to a GNSS spoofing attack. Furthermore, we demonstrate the scalability and interoperability of our algorithm through mixed-reality experiments by forming a heterogeneous MRS comprising real Crazyflie UAVs and virtual PX4-SiTL UAVs working in tandem.