By big tent I basically mean the muddied water days of there being conservative and liberal Republicans and conservative and liberal Democrats.
This is quite a challenge and potentially makes for a very intriguing timeline.
Now, Pres. Ronald Reagan was an optimistic guy. He acknowledged what could be viewed as the 'mean' side of conservatism (we're against people on welfare, we're against the Soviets), but then he takes a step beyond and emphasizes what are the positive things we're going to do instead. For example, Ronnie Reagan was a peace through strength kind of guy. Of course, the Democrats have their 'mean' side, too, as does the Reform Party, the greens, the libertarians, even the old guild socialists if you want to go way back.
Maybe if both major parties had had more of these step beyond, transcendental leaders, office holders, and candidates.
Maybe if third parties just did a better job and stirred the pot, even though usually getting well less than 5% of the vote (John Anderson and Ross Perot being exceptions).
And some specifics:
The Democrats seemed to get a reputation for being 'weak' on defense because major Democratic leaders and candidates were against the Vietnam War in '68 and '72. Maybe this could have just played out differently. Perhaps being strong defined in different but largely overlapping ways. And the Cold War in the 70s and 80s something other than just propping any old dictatorship which claims to be anti-communist.
The whole thing with the IRS vs. religious schools (most notably, the IRS vs. Bob Jones University), maybe if the persons in favor of this supposed sacred right to discriminate on racial grounds had been called out earlier, had been publicly debated more. This whole episode might seem pretty esoteric, but apparently this was a major reason why evangelical leaders became disenchanted with Pres. Carter. And many of the rank-and-file evangelicals slightly later parted company with the Democrats over abortion in '78, '79, and 1980.
Maybe if Republicans had gotten ahead of the curve on civil rights, instead of the OTL where Pres. Reagan was actually caught off guard by this very issue of Bob Jones University vs. the IRS, to the point where he felt he had to reverse the policy of his administration.
So, this would be a timeline where both parties are more centrist and where both parties compete more competently and skillfully for the center.
But perhaps you're looking for something different. There's always multiple issues cutting across each party where both are at best somewhat loose coalitions. Maybe something more like this?