Hampshire 231 and 181 for 4 (Carberry 79) require another 197 runs to beat Sussex 444 (Brown 144*, Wright 96) and 164 (Wright 61, Tomlinson 4-37)
Scorecard
Hampshire batsman Michael Carberry scored 79 to set up a tight finish against Sussex on an exciting third day at the Ageas Bowl.
Sussex were bowled out for a miserly 164 - after earlier deciding against enforcing the follow-on - to set Hampshire an improbable 378 for victory.
But Hampshire ended the day on 181 for four with 197 required to win on the final day, thanks in part to Carberry's gung-ho 86-ball 79.
Sussex cricket manager Mark Robinson said: ``We decided against the follow-on as the wicket was wearing and quite a bit of rough was beginning to appear. There was a thought it was going to get more up and down as it wore on. You are looking to bat them out the game but unfortunately what we ended up doing was give them some hope."
Carberry and Jimmy Adams set about the colossal total at a steady four an over, both crashing boundaries under the solid blue sky - putting on 76 for the opening stand.
Tymal Mills, on his first outing after moving from Essex, bowled with extreme pace and deserved the wicket of Adams, sending the Hampshire captain's off-stump sprawling.
Left-hander Carberry looked on top form as he crashed a hooked six and a cut four to bring up his 54th first-class half-century from 63 balls.
The England man had a let off when he mistimed a Mills bouncer but the ball slipped through a diving Ajmal Shahzad's hands, before producing a Gordon Greenidge-esque one-legged pull for four.
Carberry was unlucky to depart when Joyce stuck out his right paw at slip to cling on to a clipped late cut - the batsman falling for an 86-ball 79 including 13 fours and a six.
Vince came and went for nine before Liam Dawson somehow managed to play on, as the ball dramatically spun back and flick the bail off - both falling to Shahzad late in the evening.
Earlier, Hampshire concluded their first innings in the morning as Joyce ran out Tomlinson with a direct hit and left Sean Ervine unbeaten on 66.
With the home side trailing by 213 it proved a shock decision when Sussex skipper Joyce decided against enforcing the follow-on.
The 32 year-old found Wells' edge which flew to James Vince at third slip - after putting on 24 for the first wicket - and then Michael Yardy feathered the next delivery. Danny Briggs produced a brilliant catch to see off Joyce.
Nash looked settled after he scored 21 from 48 balls but left-armer Chris Wood managed to get him lbw. Craig Cachopa and first-innings centurion Ben Brown fell soon after.
Hampshire looked set to skittle through the Sussex order, leaving them 71 for six, but a 73-run partnership between Wright and Shahzad stalled the collapse.
Sussex batsman Wright scored a fine second 50 of the game after scoring 96 in the first innings, with a single after 71 balls.
Shahzad departed in the 45th over when he swept to Will Smith at silly mid-on, before Briggs found the edge of Wright's bat when he came bouncing down the track - the 30 year-old departing for 61 from 87 balls.
Mills swung across the line to give Dawson his first wicket of the season before Briggs bowled Matt Hobden - to give him match figures of 7 for 110.
A positive result is almost assured on the final day, as both Hampshire and Sussex will feel like they could win with the former still requiring 197 with six wickets remaining.
