In this paper, we study a new kind of pilot contamination appearing in multi-operator reconfigurable intelligent surfaces (RIS) assisted networks, where multiple operators provide services to their respective served users. The operators use dedicated frequency bands, but each RIS inadvertently reflects the transmitted uplink signals of the user equipment devices in multiple bands. Consequently, the concurrent reflection of pilot signals during the channel estimation phase introduces a new inter-operator pilot contamination effect. We investigate the implications of this effect in systems with either deterministic or correlated Rayleigh fading channels, specifically focusing on its impact on channel estimation quality, signal equalization, and channel capacity. The numerical results demonstrate the substantial degradation in system performance caused by this phenomenon and highlight the pressing need to address inter-operator pilot contamination in multi-operator RIS deployments. To combat the negative effect of this new type of pilot contamination, we propose to use orthogonal RIS configurations during uplink pilot transmission, which can mitigate or eliminate the negative effect of inter-operator pilot contamination at the expense of some inter-operator information exchange and orchestration.