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My dad has these external home tweeters hooked up to his stereo from Radio Shack.
By "external", you mean these are in little ~3" square boxes that sit on top of your regular speakers? I vaguely remember something like that in the Radio Shack catalog.
If you hook-up a regular (dynamic*) tweeter in parallel with your full-range speakers,
you must use a "crossover"** capacitor in series with the tweeter. Otherwise, the low frequencies will burn-it up! (Those add-on tweeters probably had an internal capacitor.)
Of course, you can never tell how any tweeter will sound with your existing speakers until you try them.
Parts Express has a
chart to help you select a crossover capacitor. It would be best to order a small selection and experiment. A smaller capacitor will push-up the crossover frequency. A larger-value capacitor will bring-in some mid-frequencies. If the crossover frequency is too low, it may sound "harsh". A very small capacitor can push the crossover point up where it becomes a "super tweeter" operating only at very high frequencies. This should give a more subtle effect. A higher frequency will also give you more power-handling, as more frequencies are blocked. (And, if you use a tweeter at lower-than-specified frequencies, it can't handle it's rated power.)
Another option is a
piezo tweeter (
example) which naturally presents a capacitive load. You can get horn versions that look OK without a box. (I have a pair, and I put some pinstripes inside the horn to improve the appearance.) They are generally very affordable, sensitive (loud), and can take high-power... (Your Radio Shack tweeters may have been piezos.)
You may need to add an R/C (resistor/capacitor) network to improve the sound. I found some information
here about making a network for a piezo tweeter. You can use a resistor network to decrease the volume level, or an R/C network to push-up the crossover frequency.
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I have a JVC home theater receiver and I noticed that I can assign highs to one speaker, mids to another, ect. Would it matter in this case what the correct phase is?
In that case, you don't have to worry about the crossover issues. But,
are you sure about that? This kind of set-up is called "bi-amped" or 'tri-amped", etc. Since a "home theater" system is usually 5.1 channels, this would require 11 amplifier-channels for a bi-amped system, and 16 amplifier-channels for a tri-amped system! It usually requires an electronic crossover and external amplifiers. Except for a separately amplified subwoofer, this type of set-up is very unusual for a home-theater or home-stereo. It is more common with high-end car audio, and very common with professional live-concert sound.
Usually, the phase is marked with +/- or coded red/black on both the speakers and the amplifier. Red is positive and black is negative (or common-ground).
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I'm on a limited budget...
Almost
everybody's on a limited budget... It's just that some people are limited to $10 and other people are limited to $1000.

* A regular-dynamic tweeter has a voice coil. A Piezo tweeter does not.
** An additional-external capacitor does not make a "proper" crossover network, but you can usually "get-away" with it. It does block the lows from the tweeter, but it does not block or "re-route" the highs from the main speaker. This presents more of a load to the amplifier (lower resistance) at high frequencies.
P.S.I found a few add-on/external tweeters:
HiVi TN28 Pyle PAHT6 Pyramid PMBHTW4 Gem Sound ST21 PowerAcoustik NX10