When the same engine was fitted to the XJ12 and later to the XJ-S, Jaguar apparently decided a side-flow radiator would be an improvement.  But what to do with those two thermostat outlets?  There are lots of potential solutions, but the engineers at Jaguar chose a terrible one.  The coolant from the left bank goes into an end tank on the left side connected to the upper 1/3 of the core.  The coolant flows left-to-right into a full-height end tank on the right side.  The coolant from the right bank enters here and mixes with the already-cooled coolant from the left side.  Then all the coolant flows right-to-left through the bottom 2/3 of the core to an end tank at the bottom left and from there to the water pump inlet.  This is referred to as the 1-1/2-pass scheme.

Does this work?  Yes, but poorly.  The left bank will always run a little warmer than the right bank because the backpressure in the upper left end tank simply must be higher than the backpressure in the right end tank; otherwise there wouldn't be any flow in the upper 1/3.  This doesn't matter a whole lot, though, until the radiator starts to get plugged up.  Pluggage tends to accentuate the problems, causing the left bank to run hotter and hotter.  Meanwhile, the sender for the temp gauge is in the right bank thermostat housing, so the driver is blissfully unaware that the left bank is overheating.

One solution:  Keep the radiator scrupulously clean.  This works, but it cannot be accomplished following Jaguar scheduled maintenance procedures.  If the mechanic follows the Jaguar book, the coolant is changed periodically and two cans of Barr's Leaks are added each time.  That crap not only plugs the radiator core, but it won't drain either, so each change just adds more and more crap to the system.

So, as a minimum:  quit using Barr's Leaks.  Better: add coolant filters.

Best: replace that 1-1/2-pass radiator with a single pass radiator.   BeCool makes an aluminum one to fit.  Actually, they make four:  62082 is their basic model, 63082 is the same thing with a nice aluminum polish, 60082 is for cars with manual transmission (no transmission cooler in the right tank), and 61082 is the manual transmission version with the polishing.

Also excellent:  John's Cars offers two different radiators intended for use in an XJ-S or XJ6/12 fitted with a Chevy engine.  Their "heavy duty" is intended for use with a small block Chevy while their "super duty" is intended for use with a big block.  They are both single-pass, and either would be suitable for use with the Jaguar V12.  Note that John's Cars also offers 1-1/2-pass radiators for use with the Jaguar V12 engine, and you will have to clarify that you don't want one of those!

You might be able to use the radiator from a pre-1988 XJ6.  Since the body is the same between an XJ6 and an XJ12, it should fit an XJ12.  And owners have reported that the radiator from a Series I/II/III XJ12 will fit an XJ-S, so the XJ6 radiator should fit both.  Another possibility is the radiator from the 6-cylinder XJ-S, which should fit the Series I/II/III XJ12 for similar reasons.  You should check the core of the radiator to ensure that it has similar cooling capacity to the V12 radiator, though; since the 6-cylinder engines are smaller, some radiators may have been constructed with fewer rows of tubes.

Finally, you can simply modify the existing radiator.  Some have two distinct tanks at the left side, while others have a single tank with a baffle hidden inside.  The latter type is easier to modify: remove the baffle (usually accomplished by a radiator shop while the tank is off for radiator service) and seal off the hose connection at the upper left.  It's preferable to enlarge the hose connection at the upper right while you're at it, but some have reported just using the 1-1/4" fitting for the flow from both banks without any problem.

Of course, once you have a single-pass radiator, you have to figure out how to connect the two thermostat outlets to a single radiator inlet.  Byrnal Haley asked his radiator man what to do, and the guy slapped this thing together from stock parts:

We should all have radiator guys that do this level of work!  The black enamel is an especially nice touch.

Haley says he used some 1-1/4" copper pipe along with several short pieces of radiator hose to put all this together.  And while many owners still watch their temp gauge obsessively, Haley can now forget about it; it stays cool no matter what.

The fitting and temp switch are to control electric fans that Haley was installing at the same time.  You can review several such installations on this site, although Haley's isn't among them.

BeCool offers an aluminum "Y-pipe" at substantial extra cost.

Joe Bialy converted his car to a single-pass system, and posted pictures of his work.  "Its the stock radiator.  I extricated the baffle plate and sealed off the upper left hose port to make it 1 pass.  I added a drain on the bottom of the right side tank and added a thermal well on the bottom of the left tank for the temperature controller.

"There are a pair of GM electric fans in there, hard to see, but they're in there.  Each is wired independent from the other for a redundant backup.

"I don't recall why I changed my banjo bolt; I did that many years ago.  What you see is a 1/4'' tube X 1/8'' NPT brass elbow fitting that has been filed down as small as I could make it because there is little room between the hood and the radiator for anything tall.  I have the dimple in the hood to prove it :(

"The air bleed has been adapted to pick up the top of the new cross over pipe.  There is also a new check valve in  the bleed hose too.

"The relays that were on the top radiator support, as well as the relays for the two fans are all located in the back left corner of the engine bay, near the left hood latch.

"Yesterday, we were stuck in traffic, a guy on a crotch rocket killed himself in a bad wreck.  The car idled in ''D'' with the A/C on max for a long time.  No temperature problems at all, no voltage problems at all, it ran just like a car would be expected to.  After all these years it's finally areliable ride."

Another idea is to use a single-pass radiator with two inlets.  This would avoid the need for a "tee" for connecting both thermostat outlets into a single radiator inlet.  Unfortunately, none of the available radiators are constructed this way.  If your radiator is brass, it's a fairly simple modification for a radiator shop to install a second inlet, or possibly to replace one large inlet with two smaller inlets.  Make sure they are spaced far enough apart to get the hoses connected.