Backscattering tag-to-tag networks (BTTNs) represent a passive radio frequency identification (RFID) system that enables direct communication between tags within an external radio frequency (RF) field. However, low spectral efficiency and short-range communication capabilities, along with the ultra-low power nature of the tags, create significant challenges for reliable and practical applications of BTTNs. To address these challenges, this paper introduces integrating an indoor reconfigurable intelligent surface (RIS) into BTTN and studying RIS's impact on the system's performance. To that end, we first derive compact analytical expressions of the probability density function (PDF) and cumulative distribution function (CDF) for the received signal-to-noise ratio (SNR) at the receiver tag by exploiting the moment matching technique. Then, based on the derived PDF and CDF, we further derive analytical expressions of outage probability (OP), bit error rate (BER), and average capacity (AC) rate. Eventually, the Monte Carlo simulation is used to validate the accuracy of the analytical results, revealing that utilizing RIS can greatly improve the performance of BTTNs in terms of AC, BER, OP, and coverage region relative to traditional BTTNs setups that do not incorporate RIS.