Underwater wireless sensor networks (UWSNs) are a prominent research topic in academia and industry, with many applications such as ocean, seismic, environmental, and seabed explorations. The main challenges in deploying UWSN are high ocean interference and noise, which results in longer propagation time, low bandwidth, and changes in network topology. To mitigate these problems, routing protocols have been identified as an efficient solution. Over the years, several protocols have been proposed in this direction and among them, the most popular are the ones that use multi-path propagation. However, there is a lack of compilation of studies that highlight the advancement of multi-path routing protocols of UWSN through the years. Hence, getting a heuristic idea of the existing protocols is crucial. In this study, we present a comprehensive survey of UWSNs multi-path routing protocols and categorize them into three main categories; energy-based routing protocols, geographic information-based routing protocols, and data-based routing protocols. Furthermore, we sub-classify them into several categories and identify their advantages and disadvantages. In addition, we identify the application of UWSN, open challenges and compare the protocols. The findings of our study will allow researchers to better understand different categories of UWSN multi-path routing protocols in terms of their scope, advantages, and limitations.