My Clash history:

I've been an active listener since I was 13, and had heard their stuff long before that due to my Dad. Now years later, in the last few weeks I have been listening to them again. Back then I had no reason to really get their albums since my Dad had them all (even now he recently got the new London Calling vinyl), but I bought the remastered London Calling and the UK debut album for myself.

Now I have the following remastered albums:

Give 'Em Enough Rope (had never heard this one before for some insane reason)
U.K. Debut
London Calling
Sandinista
Combat Rock.

So basically is this what I need (aside from Cut the Crap)?

Super Black Market Clash
U.S. Debut Album
Story of the Clash
London Calling 3 disc set

And potentially the live albums. I've only heard From Here to Eternity, how are the rest?

I'm confused between the distribution of tracks that was left off of Black Market Clash, and what kind of overlap there is. Can anybody clarify this for me?

I think you need both Black Market Clash and Super Black Market Clash. Off the top of my head, the version of "Capitol Radio" on Super BMC is the newer version they did as a bonus to their fan club or something---it's more polished and not as good as the original, IMO. The version of "Pressure Drop" is different on the two releases, and I prefer the one on regular BMC. Also, I want to say there are different versions of "Armagideon Time" and "Bankrobber" on both releases. There's a lot more stuff on Super BMC, like remixes of "Radio Clash", "The Call Up" and "Rock the Casbah". I think they're both essential. Even though it's not an official release, the regular BMC is my favorite Clash album. And as an aside, I read that that's Don Letts on the cover, who was a pioneer in bringing dub and reggae into the punk scene, and was a filmmaker/video guy, and later formed Big Audio Dynamite with Jones.

I used to have this official EP cassette with 4 mixes of Radio Clash---don't know if I still have that. I'm not familiar with some of the later compilations---is the original Radio Clash on any of them?

I'd skip Story of the Clash & instead go for Clash on Broadway. It's got some demos, live tracks, unreleased songs, and outtakes that (I don't think) have been released on anything else.

Plus it's got an unedited version of Straight to Hell that runs a couple minutes longer and has some extra verses & whatnot.

This

You also need "Rat Patrol From Fort Bragg". It's the original mix of "Combat Rock" but also includes some extra unreleased tracks. You can find it all over the web for download, as it was never released.

And if you're into bootlegs, this is a must-visit ... http://homepage.mac.com/blackmarketclash/

Forgot another one - Ellen Foley's album "Spirit Of St. Louis" features a band comprising of Strummer, Jones, Simonon and Headon (with 6 of the 12 tracks being Strummer/Jones compositions). So it's not really a Clash album, but it kinda is.

And here's a clip of Strummer and Jones playing together for the last time, just 5 weeks before Joe died ... "White Riot" and "London's Burning".

<iframe width="420" height="315" src="http://www.youtube.com/embed/hu-02udB57w" frameborder="0" allowfullscreen></iframe>

I don't think you need the US debut if you've got the UK debut and get the singles collection. Maybe the other collections make the singles collection superfluous? I'm not sure.