Integrity monitoring is an essential task for ensuring the safety of positioning services. Under a selected probability of hazardous misleading information, the protection levels (PLs) are computed according to a considered threat model to bound the positioning errors. A warning message is sent to users when the PL exceeds a pre-set alert limit (AL). In the short-baseline real-time relative kinematic positioning, the spatially correlated errors, such as the the orbital errors and the atmospheric delays are significantly reduced. However, the remaining atmospheric residuals and the multipath that are not considered in the observation model could directly bias the positioning results. In this contribution, these biases are analysed with the focus put on the multipath effects in different measurement environments. A new observation weighting model considering both the elevation angle and the signal-to-noise ratios is developed and their impacts on the positional results are investigated. The coefficients of the proposed weighting model are determined for the open-sky and the suburban scenarios with the positional benefits maximised. Next, the overbounding excess-mass cumulative distribution functions (EMCs) are searched on the between-receiver level for the weighted phase and code observations in these two scenarios. Based on the mean and standard deviations of these EMCs, horizontal protection levels (HPLs) are computed for the ambiguity-fixed solutions of real experiments. The HPLs are compared with the horizontal positioning errors (HPEs) and the horizontal ALs (HALs). Using the sequential exclusion algorithm developed for the ambiguity resolution in this contribution, the full ambiguity resolution can be achieved in around 100% and 95% of the time for the open-sky and the suburban scenarios, respectively. The corresponding HPLs of the ambiguity-fixed solutions are at the sub-dm to dm-level for both scenarios, and all the valid ambiguity-fixed HPLs are below a HAL of 0.5 m. For the suburban scenario with more complicated multipath environments, the HPLs increase by considering extra biases to account for multipath under a certain elevation threshold. In complicated multipath environments, when this elevation threshold is set to 30 degrees, the availability of the ambiguity-fixed solutions could decrease to below 50% for applications requiring HAL as low as 0.1 m.