Service for imported iPhone I bought a grey-market import iPhone 7 plus in the Phillipines. It's a Hong-Kong phone, per the authorized agent in Manilla.

How can I get it serviced by Apple, as the authorized agent doesn't want to cover the work, given it's import status.
It sounds to me like the store you bought it from, imported it as a grey import from Hong Kong. However, all iPhones come with a global warranty, so you will be covered. 

You need to contact Apple directly - someone there will be able to organise something for you. You may need to provide proof of purchase, so have that ready (scan it in if you haven't got a digital copy).

For your info, below is a similar experience I'm aware of firsthand.

I recently helped a lady who purchased a new iPhone from a local online store. About 10 months later the iPhone developed a fault, and while it was definitely under warranty, it took a while to establish this fact because the serial number came up as having been purchased in Thailand, and because it was dated about 7 weeks earlier than when she had purchased it. 

In this case, she needed to show her purchase receipt to correct the commencement date for her warranty, as it was originally showing as only 8 days remaining compared to the two months she thought she had left. It did get all sorted, but the issue had to get escalated before it was resolved.