Short answer: No. The IRS says TRICARE coverage does not meet the HDHP deductible rules, so a person covered by TRICARE is not an HSA-eligible individual, even if they are also on an HDHP.
IRS Notice 2004-50 (Q&A-6) answers this directly: coverage options under TRICARE do not meet the minimum annual deductible requirements for an HDHP, so an individual covered under TRICARE is not an eligible individual and may not contribute to an HSA.
This catches military retirees and reservists who join an employer’s HDHP while keeping TRICARE (including TRICARE Reserve Select). If the employer makes HSA contributions for an ineligible employee, those contributions have to be corrected.
The test looks at each person’s own coverage. An employee whose own coverage is only the HDHP is not disqualified just because a family member has TRICARE, but an employee who is personally covered by TRICARE is. Veterans using VA care follow a different rule (see the separate VA and HSA card).