Here's a quick write-up on one way to convert your single filament (1056 bulb) turn signals into dual-filament units that use 1157 bulbs, giving you running lights, and brighter turn signals. The 1056 bulb is 23 watts, while the 1157 is 4 watts on low filament, and 27 watts on high.
For each signal, you'll need a two-contact socket. I used a Motormite/Conduct-Tite 84809 socket, designed for 1970-1994 Subaru, Toyota and Acura vehicles. They are available at Pep Boys, and other auto parts stores. The socket is encased in a plastic mount, and doesn't look useful at first; however, the socket can just be pushed out of the housing, as shown below. Discard the plastic housing and the ground wire.
Rick K. suggests going to an automotive salvage yard, i.e. Pick A Part; the taillights from '70s and '80s Datsuns and Toyotas should be ready to use, and you can get them for as little as $1/six (with bulbs!).
Socket and packaging
Push socket out of plastic housing
Remove and disassemble your stock turn signal. You will wind up with a socket attached to an angle bracket, as shown below. The socket is attached to the bracket via two spot welds. Drill out the welds to remove the original socket. (Just drill far enough to break the socket free. If you have drill bits with a center "brad point" for starting, these work well.)
Stock socket and bracket. One of the spot welds is visible.
Now, you can install the replacement socket. First, you want to set up the alignment of the socket in the bracket. Ideally, the filaments should be horizontal, so install a bulb in the socket, and stick the socket in the bracket. The bracket extends horizontally, so align the filaments and make a reference mark with a magic marker on the socket and bracket. Remove the bulb.
There are two choices for mounting the socket in the bracket, welding or epoxy. I have access to a welder, so I went that route. I also wanted to use the bracket for the ground, as the new socket doesn't have a ground tab welded onto it like the stock unit. If you epoxy the socket, you will have to attach a ground wire either by attaching it to the spring, or soldering it to a copper washer and putting that below the spring. (I'll try to find a picture of the latter.) Either way, you'll want to remove the plastic contact holder before attaching the socket, as you don't want to melt it or lock it up with epoxy. For depth, the open edge of the socket should line up with the cupped edge of the bracket. (Look at the stock unit before you take it apart to get an idea of that.) Also keep the welding or epoxy away from the slots in the socket. If you weld the socket in, you will probably need to smooth up the inside of the socket with a Dremel tool once you are done.
Bracket with new socket welded on
With the new socket installed, replace the plastic contact holder, spring, and wires/contacts. The wires that come with the socket are not long enough, and will have to be extended. You can avoid doing one of them by replacing it with the stock wire. Just pull the aftermarket wire out, and stick the stock one in. Also make sure that the wires are in the right holes. This is easiest to check by placing a bulb in the socket and using a battery or ohmmeter to see which wire goes to which filament. The bright/turn filament should go to the grey wire, and the dim/marker filament should go to red. For the ground, I crimped a 3/8" ring terminal on to the black wire, and placed it on the turn signal stalk, so the bracket nut would hold it down. I'll try to get a picture of that.
Socket assembly with wires installed.
If you use 18-ga wire for the marker light (20 is probably OK too), you should be able to thread all three wires through the turn signal stalk. You will have to get all of the wires straight, and stacked in a triangle, so the wires don't twist around each other. If they twist, the bundle will be too bulky to fit through the stalk. Run the wires, put the rest of the socket parts back together, and tighten the nut down. Note that the rubber grommet in the signal side of the stalk won't fit all three wires, so I used a piece of shrink tube there to provide chafe protection. Hook up the marker light wire to an ignition-on power line, and you're all set. I ran mine through a switch on the mini-dash so I could turn them on as needed, and conserve power for the off-road lights on the trail. I will try to get some pics of the final re-assembly.
If someone in the LA area wants to convert their signals, and has a digital camera, we can get some more extensive pics.
Note that the 1157 bulb draws an extra 4W over stock on the turn signal filament. This will cause your flasher to go faster than it should. The problem can be fixed by using a "universal wattage" flasher, which is around $10, also at auto parts stores. I used a Tridon Stant brand electronic universal flasher, #EP34. This is a three-terminal unit that handles any wattage bulb, and gives a "bulb out" indication by flashing at double speed. The stock harness plugs into two of the terminals, and you simply need to connect a ground wire to the third. I'll post the wire colors once I look them up. Note that the terminal diagram on the package shows the socket, not the flasher. There should be an accurate diagram printed on the flasher itself.
DISCLAIMER: I did all four turn signals on my KLR, and the rear ones show up especially well at night, as they are the same brightness as the taillight. (The fronts get somewhat lost in the headlight glare.) However, it might be (is probably) illegal to run amber running lights in the rear, so "check local laws before installing," as they say. I haven't gotten pulled over yet, and as a fellow KListeR said, "If I get pulled over, at least I know they can see the lights." One remedy would be to find red lenses for the rear, as cars used to have red turn signals, so that should be legal. However, I don't think the signals will be as noticeable as the amber ones.
A couple of KLR list members had ideas on coloring the rear lenses red. Rick K. found a water-based tint paint in a hobby shop that has held up for two seasons so far. George B. from Escondido also recommends Dekka, which is a simulated stained glass product available from art supply or stained glass shops. Crimson #29 is the one you want.