We already knew that names were problematic but I agree yours will be a source of confusion.
I wouldn't know where to separate the names, if you write name(s) first or the surname(s). Not even the adequate pronoun.
Stating the fields so explicitly is uncommon, but seems the most sound solution. I would probably try to make it blend more with the rest of the signature, by also naming other fields, like an id card.
Additionally, I would recommend to sign with the name you would like to be addressed. Some people disagrees on this, but it's a hint that can help convey ho you want to be addressed. So, you could end the email as
Sincerely, Man Ka if you wanted to be addressed as
Dear Man Ka.
Sincerely, Mr Au Yeung is not something I have seen, although it's used with other prefixes such as
Sincerely, Dr Au Yeung.
Using a non-breaking space between the parts, i.e.
Man Ka Au Yeung may help for computers that automatically split on a space character, but is way too subtle for humans.
Man Ka, case Specialist
Elm Trade Inc.
First name: Man Ka
Last name: Au Yeung
Telephone: +12 1234 5678
Fax: +12 1234 5678
Address: Street, Floor, Borough, City, Province
Postal code: 12345
Goal: serve the needs of our customers as if they were our own ones
Ultimately, it's your choice how to present a signature. If this is for a corporate email, the company may have some guidelines (or even hard rules) on what and how to set. Otherwise, it is up to you what and how to say, depending on what you want to convey (which is not just the text itself). Signatures range from long and extremely formal to signing with emoji, such as Senator 🐮👑. In some cases the goal could be a "dull" signature to be perceived as seriousness, whereas a startup could prefer to emphasize instead a lightly style more familiar, or use made-up titles. Some will include complex images in the signature that take ten times the space of the actual email body, others will go for simplicity and use a plaintext one. Listen to feedback you may get, of course, see how you see your own signature as you use it, and tweak as many times as needed, but there's not a "wrong" signature.¹
¹ When talking about the body itself. I'm not including there technical errors (such as malformed html), orthographic mistakes, or technology choices. Only the high-level definition on what to write.