Gentlemen,
I installed a 140 ft wire in hopes to match it on all HF bands. It
plays nicely through my ATU. I would like to modify this setup by
using about 20 ft.of coax inside the house (for a better look). Is
there a known matching scheme to accomodate the wire to the coax? What
other problems might I encounter?
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The known matching scheme is a tuner at the coax/antenna wire junction. For
multi-band operation a balun is fairly useless.

But if you don't fancy walking 20 feet to re-tune every time you change
frequency, reading the SWR meter with a telescope, why not do what I do. I
just extend the antenna wire into the house using plastic insulated 18 gauge
wire or something similar.

On the 160 and 80 meter bands you will notice no difference in performance
or behaviour. Just a difference in tuner settings. On the 15 and 10 meter
bands there might be an increase in RF inside the house over what you
usually get. You always get SOME RF in the shack/house with end-fed
antennas.

For an endfed wire you need a good ground near to the end of the antenna.
When the end of the antenna is in the house the domestic plumbing system
makes a very good ground. The hot and cold metal water pipes, with pipes to
central heating radiators in each room, constitute an excellent set of long
and thick "radials". It would be a great pity not to make use of them.
Make 1, 2 or perhaps 3 connections from the transmitter to different nearest
water pipes. Unobtrusive 18-gauge bare wire under carpets is fine. The
transmitter will also pick up an RF ground via the 110V or 230V power
wiring.
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Reg, G4FGQ