Backscattering tag-to-tag networks (BTTNs) are emerging passive radio frequency identification (RFID) systems that facilitate direct communication between tags using an external RF field and play a pivotal role in ubiquitous Internet of Things (IoT) applications. Despite their potential, BTTNs face significant security vulnerabilities, which remain their primary concern to enable reliable communication. Existing authentication schemes in backscatter communication (BC) systems, which mainly focus on tag-to-reader or reader-to-tag scenarios, are unsuitable for BTTNs due to the ultra-low power constraints and limited computational capabilities of the tags, leaving the challenge of secure tag-to-tag authentication largely unexplored. To bridge this gap, this paper proposes a physical layer authentication (PLA) scheme, where a Talker tag (TT) and a Listener tag (LT) can authenticate each other in the presence of an adversary, only leveraging the unique output voltage profile of the energy harvesting and the envelope detector circuits embedded in their power and demodulation units. This allows for efficient authentication of BTTN tags without additional computational overhead. In addition, since the low spectral efficiency and limited coverage range in BTTNs hinder PLA performance, we propose integrating an indoor reconfigurable intelligent surface (RIS) into the system to enhance authentication accuracy and enable successful authentication over longer distances. Security analysis and simulation results indicate that our scheme is robust against various attack vectors and achieves acceptable performance across various experimental settings. Additionally, the results indicate that using RIS significantly enhances PLA performance in terms of accuracy and robustness, especially at longer distances compared to traditional BTTN scenarios without RIS.