Opener Zakir Hasan was unbeaten on 16 alongside Mominul Haque, who has yet to score, as Bangladesh built a slender lead of 30 runs after Mahmudul Hasan Joy and skipper Najmul Hossain Shanto departed in fading light.
New Zealand spinner Ajaz Patel struck with his third ball before captain Tim Southee dismissed Shanto in the eighth over, leaving the hosts in some trouble before play was stopped and eventually called off.
An aggressive Phillips earlier smashed nine fours and four sixes during his knock of 87 as New Zealand made 180 all out in their first innings for a lead of eight runs.
The Black Caps were 55-5 after the first day, having bowled out Bangladesh for 172 before rain washed out the second day at Mirpur's Shere Bangla National Stadium but they launched a solid counter-attack thanks to Phillips' career-best knock.
After the start was delayed due to overnight rain, Phillips took on the Bangladesh spinners during a 49-run partnership with Daryl Mitchell before turning aggressor again in a 55-run stand with Kyle Jamieson.
Mehidy Hasan Miraz finished with figures of 3-53 and Taijul Islam claimed 3-64 with Shoriful Islam and Nayeem Hasan picking up two wickets each to dismiss New Zealand in 37.1 overs on a track that got tougher for batting.
Bangladesh are looking to complete a sweep after winning the opening match of the two-test series by 150 runs in Sylhet.
