The bullpen has probably been the best aspect of the team this month. They've just been pressured sometimes because the other aspects can't put away games.
Also, for all the talk of not having a closer, we have a closer. 9 of Neris's 10 appearances in May have been finished games (including some that started in the 8th). No blown saves all year. In May, Neshek had two saves and Nicasio one. Seranthony has only pitched the 9th twice all year (one non-save situation and one game in which the Phillies trailed).
As a team, the Phillies only have 5 blown saves, 3rd-fewest in the NL, and those aren't all 9th inning, or losses. The most recent was Seranthony's 7th/8th against the Cubs.