We present an anytime Monte Carlo tree search (MCTS) algorithm to generate real-time, near-optimal search paths in large subsea environments. The MCTS planner continuously builds a tree of the search space until either the allowed time per move is reached or the budget constraint for the search mission is met. In order to improve the performance of the MCTS planner, we propose a novel heuristic action selection policy to determine the value of a leaf node. The proposed heuristic is tailored to problems where making a turn incurs a higher cost than moving straight, such as the case on autonomous underwater vehicles. Through extensive simulations, we show that our heuristic yields a significant performance improvement over a lawnmover path planner - a commonly employed approach in subsea search applications - and over a simple MCTS planner where actions are selected uniformly at random. In our numerical illustrations, we use a real data set abstracted from sonar measurements acquired from the Boston Harbor.