An obvious fake?  

I think it's 100% real
When it was initially produced, it was available in white, as well as black. I don't know when HM started to offer it in naked metal.
Congrats...it's a beauty.

More photos...
I've not seen one of these table bases ever disassemble the way this one does. The pieces don't even resemble one solid piece. And also, the tabletop looks to be some kind of fiberboard with veneer stripping around the sides.

So original then, barry?
The base paint is in bad shape. But the fellow told me that in advance, so I was already planning on powder coating it. The assembly looks right to you though? Have a look at where the tube meets the pedestal base. Pretty rough.

its real
its real

Your photos look like it's in better
condition than you say it is. I'd just live with it.
I have a pair and both of the table tops are black and both of the bases are the aluminum metal color, but one is shiny, glossy and the other is dull. Both look like that's the way they should be, but since I don't keep 'em side by side, it doesn't matter.
I prefer the dull finish, but if you bought a new one, it would be glossy shiny.

Lunch
When did you get that table? Looks good to me. I don't remember the base coming off in two, but I've never owned one either.
You should sell it to me.
Looks like your floors came out dandy.
My folks have a bikini pad waiting for you. Kiss dad on the mouth for me 😉

barry...
Not trying to make it sound worse than it is. I may leave it as is. Normally, internet photos don't catch everything. The top of the pedestal feet is where the paint is chipped most. There is a bright side to this, however. Apparently, this table is older than any I've seen. Like whitespike, I've never seen one that disassembles this way. Not unhappy with it. Just wanted to ensure its authtenticity. Thanks for the input. And thanks to LRF as well.
whitespike, floors came out great. Bought this table some time in October I think. Reminds me of an ice cream sandwich. And I will thank your father for the pad exactly as you suggest...

i have both tables and ...
i have both tables and the white one is chipped just like yours.

I have 4 newer versions of...
I have 4 newer versions of the coffee tables and the base doesn't come apart like that, but the press-board edge looks right to me...that's how mine are.
fm

Real deal
The reason the base is segmented is because they offered it in two heights. The top and bottom of the base are the same, the difference in height is created with a shorter center pole.

Is there more?
Are you showing all the parts to your table? I ask because there should be a long threaded rod that has washers and bolts on the top and bottom. That rod holds the 3 pieces of the base together. Typically rather than dismantling the entire base, you leave the base intact and just remove the top (with the 4 screws you can see holding it on) to ship it safely.
I would not paint the base either. I have a couple of these, and really like the patina of age on the bases.
Also, your round medalion dates the table to the 1960s or early 70s, but there is an earlier version of this base design. The foot portion of the base turns down to the floor rather than following the fluid line coming from the center. It's a subtle detail, but I like the original design.
Here are some pictures so you can see 1st generation base. Of course the base was used with the minature chests and other end tables/ ocational tables as well.

Yes, Pegboard
There's more. I didn't photograph the assembly hardware. And I am missing a nut and washer. But I'm just going to go to the hardware store and match the two I have to remedy that.
However, I think what has been bothering me with this table is deformation to both ends of the cylindrical tube portion of the base. Whereas most of these tables show a clean seam where this piece meets the tabletop and swag base, my table looks a bit ragged at both points. Looks as if someone got carried away with a ratchet wrench and bent each end of the tubing out a bit. I'll snap a closer shot of it when I get the chance. Not sure whether I want to keep the table or not at this point. Bothersome. Trite. But bothersome.