How make arpeggio span two staves?

Unfortunately you must use something hidden to make this work.
First - enter the arpeggio the way you want it to look, then select all the notes and uncheck play in the inspector (F8) so they will be silent.

Next you either need to make another staff with the same instrument (recommended if there are a lot of these in the song) or in an unused voice in the lowest clef staff make an arpeggio that uses all the notes from both staffs. If I use another voice I usually just find an empty measure somewhere and put it on the same beat as it would be in the measure I'm working on. Make sure the entire chord for all staffs (usually 2 but could be 3 like an organ) is on the lowest staff with the arpeggio symbol on it. The reason I say to use the lowest staff is that it is easier to build a chord from bottom to top. Now make this work measure invisible by pressing V (at this point you need to make sure Show invisible is checked under the view menu). In the selection filter (F9) uncheck voice 1 and select the measure. You should now have only the voice with your arpeggio selected. Cut that measure. Select the first beat in voice 1 in the measure you want to place the arpeggio and paste it. Make this a habit, because MS will complain if there is a tuplet in the measure and you select the entire measure.

If you make an invisible staff, you will have to make it visible to add notes you want to hear of course, but you can put everything into voice 1 without the need to hide anything. You can even delete all but one staff and simply put the appropriate clef on the staff as needed for arpeggios.

I do the invisible voice option often mainly because I don't like invisible staffs.

There's a trick I learned from a forum member regarding arpeggios and time-stretching, and it also applies to cross-staff or two voiced arpeggios, but it takes some calculation.

1) Make an arpeggio for both your F & G clef chords.
2) Hide one of them, the top one, and drag up the bottom one to fit
3) Looks fine, but as you said they don't sound right because they both trigger at the same instance rather than a continuous unfolding
4) Use the piano-roll editor. Find out the OnTime values for the Bass clef chord by right-clicking in the bass clef measure and opening the piano-roll editor. For a 4-tone chord of eighth notes I get the interval of 175 from the first note being 0 and the next being 175, next 350, then 525, so you want the next note, the first note of the treble clef chord to be 525+175, and then the next note of that one to be 525+175+175, etc. You have to right click into the treble clef, or whatever clef you want to be looking at and open the piano roll editor from there. Apparently it doesn't encompass both clefs in one window. One at a time click each note and then type in the numeral value for it in the OnTime form.

I hope this makes sense, I'm tired and might be a little unclear. This is definitely a work around, but it works and doesn't require some extra hidden staves or anything like that; plus, it's good to know these things anyways :)

I'm a relative newbie to MuseScore, though I've used other notation s/w before. (I'll never go back by the way. Wish I found MuseScore sooner!!) My solution to the continuous arpeggio problem (and everything I haven't yet found the "proper way" to do) is to brute-force it.

In the lower staff, I'd shorten the previous note a little (say a 32nd note) and place my lower arpeggiated chord there. Then, I the identical chord is immediately repeated, right where it should have been in the first place. Finally, I tie the 2 chords together with TIES. On playback, this should approximate the effect you need---effectively shifting the lower arpeggio a tiny bit ahead of the upper one, thereby yielding a pseudo-continuous arpeggio from bottom note to top note. (Attached: the chord in the red circle would have been a half note in length, but is cut short by 1-32nd to "make room" for the effect.)

See #312181: Need ability to extend arpeggios to other voices and staves

I believe if you put the arpeggio on the chord in the top staff and then double click the arpeggio, 3 or to little squares will pop up on the ends of the arpeggio and then you click the bottom square or dot thing and then once its selected you hold shift and then the down arrow key kind of like when you move something up or down to different staff and then it should expand down to the next staff where you want it and playback should work on it too. it must be done from top to bottom I believe though. I hope this makes sense.