(Well, it interested me, anyway)
I found this site where the guy made a 38kHz carrier wave transmitter with a 555. Nothing surprising, just a matter of speccing the components to get the frequency, comes out at 37.6kHz in fact.
But, with the components given, it’s very asymmetric as shown in the blue 'scope trace below: about 85% high (22/26). (I actually have a 4k7 instead of the 10k variable.)
Yet, as shown by the yellow trace, the TSOP goes low as it should for a 38kHz carrier. I had just assumed the carrier needed to be 50/50, but on reflection I guess there’s no reason it should be: all the TSOP looks for is the edge of the next cycle I suppose.
Next test is to find another 555 and use that to trigger the first one, to superimpose a message on the carrier. I think I should be able to that on its /reset pin.